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

1,3

Comments

An equivalent definition: take the polynomials corresponding to rows 2, 4, 6, 8, ... of A060187, divide by x+1, and extract the coefficients. [Corrected by Petros Hadjicostas, Apr 17 2020]

Links

G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

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

Example

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;
  ...

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}]]

Prog

(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

Crossrefs

Cf. A002671 (row sums), A034870, A060187, A171692, A225398.

Sequence in context: A291072 A174599 A291074 * A022185 A176631 A015150

Adjacent sequences: A225073 A225074 A225075 * A225077 A225078 A225079

Keyword

nonn,tabf

Author

Roger L. Bagula, Apr 26 2013

Extensions

Edited by N. J. A. Sloane, May 06 2013, May 11 2013

Status

approved