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A156985 Triangle formed by coefficients of the expansion of p(x,n) = (1-x)^(2*n + 1)*Sum_{j >= 0} (1 +j +j^2)^n * x^j. 1
1, 1, 0, 1, 1, 4, 14, 4, 1, 1, 20, 175, 328, 175, 20, 1, 1, 72, 1708, 9784, 17190, 9784, 1708, 72, 1, 1, 232, 14189, 199616, 884498, 1431728, 884498, 199616, 14189, 232, 1, 1, 716, 108250, 3353948, 31986447, 115907544, 176287788, 115907544, 31986447, 3353948, 108250, 716, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

FORMULA

T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1-x)^(2*n + 1)*Sum_{j >= 0} (1 +j +j^2)^n * x^j.

Sum_{k=0..2*n} T(n, k) = A010050(n).

EXAMPLE

Irregular triangle begins as:

  1;

  1,   0,     1;

  1,   4,    14,      4,      1;

  1,  20,   175,    328,    175,      20,      1;

  1,  72,  1708,   9784,  17190,    9784,   1708,     72,     1;

  1, 232, 14189, 199616, 884498, 1431728, 884498, 199616, 14189, 232, 1;

MATHEMATICA

p[x_, n_] = (1-x)^(2*n+1)*Sum[(1+k+k^2)^n*x^k, {k, 0, Infinity}];

Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten

PROG

(Sage)

def T(n, k): return ( (1-x)^(2*n+1)*sum((j^2+j+1)^n*x^j for j in (0..2*n+1)) ).series(x, 2*n+2).list()[k]

flatten([1]+[[T(n, k) for k in (0..2*n)] for n in (1..12)]) # G. C. Greubel, Jan 07 2022

CROSSREFS

Cf. A010050, A156896, A156890, A156901, A156918.

Sequence in context: A107775 A003117 A239465 * A138229 A131702 A276826

Adjacent sequences:  A156982 A156983 A156984 * A156986 A156987 A156988

KEYWORD

nonn,tabf

AUTHOR

Roger L. Bagula, Feb 20 2009

EXTENSIONS

Edited by G. C. Greubel, Jan 07 2022

STATUS

approved

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Last modified May 25 20:09 EDT 2022. Contains 354071 sequences. (Running on oeis4.)