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 A155796 Triangle read by rows: first define the Narayana numbers: Y(n,m)=Binomial[n, m]*Binomial[n + 1, m + 1]/(n - m + 1); then t(n,m)=Sum[(-1)^j *Y(n + 1, j)*(k + 1 - j)^n, {j, 0, k + 1}] 0
 1, 1, -2, 1, -2, -33, 1, 1, -109, -209, 1, 11, -324, -894, 1641, 1, 36, -867, -4262, 12951, 85926, 1, 92, -2085, -20516, 74369, 625164, 1435939, 1, 211, -4419, -93989, 344617, 4306671, 9337441, -7280909, 1, 457, -7652, -402971, 1253140, 28570687 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums are: {0, 1, -1, -34, -316, 435, 93785, 2112964, 6609624, -1422008070, -59772756330,...}. LINKS Table of n, a(n) for n=0..41. FORMULA Narayana numbers: Y(n,m)=Binomial[n, m]*Binomial[n + 1, m + 1]/(n - m + 1); Eulerian sum form: t(n,m)=Sum[(-1)^j *a(n + 1, j)*(k + 1 - j)^n, {j, 0, k + 1}] EXAMPLE {1}, {1, -2}, {1, -2, -33}, {1, 1, -109, -209}, {1, 11, -324, -894, 1641}, {1, 36, -867, -4262, 12951, 85926}, { 1, 92, -2085, -20516, 74369, 625164, 1435939}, {1, 211, -4419, -93989, 344617, 4306671, 9337441, -7280909}, {1, 457, -7652, -402971, 1253140, 28570687, 62681044, -180621929, -1333480847}, {1, 958, -7325, -1618693, 2748769, 182235307, 459030881, -2310249875, -15651462716, -42453433637} MATHEMATICA A[n_, m_] = Binomial[n, m]*Binomial[n + 1, m + 1]/(n - m + 1); t[n_, k_] = Sum[(-1)^j *A[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; Table[Table[t[n, k], {k, 0, n - 1}], {n, 1, 10}]; Flatten[%] CROSSREFS Sequence in context: A281129 A136156 A191657 * A341530 A141238 A094690 Adjacent sequences: A155793 A155794 A155795 * A155797 A155798 A155799 KEYWORD sign,tabl AUTHOR Roger L. Bagula, Jan 27 2009 EXTENSIONS Edited by N. J. A. Sloane, Jan 31 2009 STATUS approved

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Last modified September 23 21:37 EDT 2023. Contains 365554 sequences. (Running on oeis4.)