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A225076 Triangle read by rows: absolute values of odd-numbered rows of A225356. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

An equivalent definition: take the polynomials corresponding to rows 1,3,5,7,... of A060187, divide by x+1, extract coefficients.

Row n (n>=1) has length 2n-1.

The row sums are A002671.

LINKS

Table of n, a(n) for n=1..34.

FORMULA

Triangle read by rows: row n gives coefficients in expansion of the polynomial ((x-1)^(2*n)/(x+1))*(sum_{k=0..oo} (2*k+1)^(2*n-1)*x^k). The infinite sum simplifies to a polynomial.

t(n,m)=2^(-1+2 n) (-1+x)^(2 n) LerchPhi (x,1-2 n,1/2)/(x+1).

EXAMPLE

Triangle begins:

{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[FullSimplify[ExpandAll[p[x, n]/(1+x)]], x], {n, 1, 10}]

(* alternative: recursive *)

l = 2; A[n_, 1] := 1; A[n_, n_] := 1;

A[n_, k_] :=

A[n, k] = (l*n - l*k + 1) A[n - 1, k - 1] + (l*k - (l - 1)) A[n - 1, k];

B[n_, k_] := B[n, k] = A[n + 1, k + 1];

Flatten[Table[CoefficientList[FullSimplify[ExpandAll[Sum[B[n, k]*x^k, {k, 0, n}]/(x + 1)]], x], {n, 1, 14, 2}]]

(* second alternative method: polynomial expansion *)

p[t_] = Exp[t]*x/(-Exp[2*t] + x); Flatten[ Table[ CoefficientList[FullSimplify[ExpandAll[(n!*(-1 + x)^(n + 1)/(x*(x + 1)))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 1, 13, 2}]]

CROSSREFS

Cf. A002671, A171692, A060187, A034870, 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

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Last modified April 20 02:06 EDT 2019. Contains 322291 sequences. (Running on oeis4.)