Rational generating functions for the Certain Stanley-Stern Sums By Shalosh B. Ekhad Theorem Number, 1 --------------------------------- Let Z[n] be the integer sequence whose generating function is infinity ----- 3 2 \ j -2 t - t + 1 ) Z[j] t = --------------------- / 4 3 2 ----- -t - t - t - t + 1 j = 0 Let n - 1 --------' ' | | Z[i + 3] (Z[i + 2] + Z[i + 3]) F[n](x) = | | (1 + x + x | | | | i = 0 (Z[i + 1] + Z[i + 2] + Z[i + 3]) + x (Z[i] + Z[i + 1] + Z[i + 2] + Z[i + 3]) + x ) Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ H(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n ) H(n) t / ----- n = 0 equals (I-Mt)^(-1) v [1] where M is a certain square matrix of dimension, 1 and v is a certain vector of length, 1 that are too big to display. At any rate we can use them to find the first, 31, terms starting at n=0, for the sake of the OEIS. [1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328125, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125, 59604644775390625, 298023223876953125, 1490116119384765625, 7450580596923828125, 37252902984619140625, 186264514923095703125, 931322574615478515625] ----------------------------- This took, 0.006, seconds. Theorem Number, 2 --------------------------------- Let Z[n] be the integer sequence whose generating function is infinity ----- 3 2 \ j -2 t - t + 1 ) Z[j] t = --------------------- / 4 3 2 ----- -t - t - t - t + 1 j = 0 Let n - 1 --------' ' | | Z[i + 3] (Z[i + 2] + Z[i + 3]) F[n](x) = | | (1 + x + x | | | | i = 0 (Z[i + 1] + Z[i + 2] + Z[i + 3]) + x (Z[i] + Z[i + 1] + Z[i + 2] + Z[i + 3]) + x ) Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ H(n) = ) a(n, k) a(n, k + 1) / ----- k = 0 Then infinity ----- \ n ) H(n) t / ----- n = 0 equals (I-Mt)^(-1) v [1] where M is a certain square matrix of dimension, 1490 and v is a certain vector of length, 1490 that are too big to display. At any rate we can use them to find the first, 31, terms starting at n=0, for the sake of the OEIS. [0, 4, 63, 868, 11233, 145036, 1896034, 24702594, 320939693, 4168509507, 54161394007, 703690375378, 9141431570591, 118745454279821, 1542455665520992, 20035967844104422, 260258398681159859, 3380598151181562105, 43911614329670982982, 570381048569505299346, 7408847766971480584012, 96235493526207790122176, 1250026616015747916526358, 16236897964945427549673250, 210904998373441401920840498, 2739495053377290644149593887, 35583941330076637030005311418, 462208066131520527815955631320, 6003727735438413886765517794080, 77983806894072688210608805413357, 1012949635313850740084133122142035] ----------------------------- This took, 188.849, seconds. ----------------------------------------- This concludes this article that took, 239.528, seconds to produce.