A225076 Triangle read by rows: absolute values of odd-numbered rows of A225356.
1, 1, 22, 1, 1, 236, 1446, 236, 1, 1, 2178, 58479, 201244, 58479, 2178, 1, 1, 19672, 1736668, 19971304, 49441990, 19971304, 1736668, 19672, 1, 1, 177134, 46525293, 1356555432, 9480267666, 19107752148, 9480267666, 1356555432, 46525293, 177134, 1
Offset: 1
Examples
Triangle T(n,m) (for n >= 1 and 0 <= m <= 2*n - 2) begins as follows: 1; 1, 22, 1; 1, 236, 1446, 236, 1; 1, 2178, 58479, 201244, 58479, 2178, 1; 1, 19672, 1736668, 19971304, 49441990, 19971304, 1736668, 19672, 1; ...
Links
- G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened
Programs
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Mathematica
(* Power series via an infinite sum *) p[x_,n_] = (x-1)^(2*n)*Sum[(2*k+1)^(2*n-1)*x^k,{k,0,Infinity}]; Table[CoefficientList[p[x,n]/(1+x),x],{n,1,10}]//Flatten (* First alternative method: recurrence *) t[n_, k_, m_]:= t[n,k,m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k - (m-1))*t[n-1,k,m]]; T[n_, k_]:= T[n, k]= t[n+1,k+1,2]; (* t(n,k,2) = A060187 *) Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k,0,n}]/(x+1), x], {n,14,2}]] (* Second alternative method: polynomial expansion *) p[t_] = Exp[t]*x/(-Exp[2*t] + x); Flatten[Table[CoefficientList[(n!*(-1+x)^(n+1)/(x*(x+1)))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 1, 13, 2}]] -
Sage
def A060187(n, k): return sum( (-1)^(k-j)*(2*j-1)^(n-1)*binomial(n, k-j) for j in (1..k) ) def A225076(n,k): return sum( (-1)^(k-j-1)*A060187(2*n, j+1) for j in (0..k-1) ) flatten([[A225076(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022
Formula
Triangle read by rows: row n gives coefficients in the expansion of the polynomial ((x - 1)^(2*n)/(x + 1)) * Sum_{k >= 0} (2*k + 1)^(2*n-1)*x^k. The infinite sum simplifies to a polynomial.
Sum_{m=0..2*n-2} T(n,m)*t^m = 2^(2*n-1) * (1-t)^(2*n) * LerchPhi(t, 1-2*n, 1/2)/(1 + t).
Sum_{k=1..n} T(n, k) = A002671(n-1).
T(n,m) = Sum_{k=0..m-1} (-1)^(m-k-1)*A060187(2*n,k+1) for n >= 1 and 1 <= m <= 2*n-1. - Petros Hadjicostas, Apr 17 2020
Extensions
Edited by N. J. A. Sloane, May 06 2013, May 11 2013
Comments