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A178619
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Triangle T(n,k) with the coefficient of [x^k] of the series (1-x)^(n+1)* sum_{j>=0} binomial(n + 4*j, 4*j)*x^j in row n, column k.
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1
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1, 1, 3, 1, 12, 3, 1, 31, 31, 1, 1, 65, 155, 35, 1, 120, 546, 336, 21, 1, 203, 1554, 1918, 413, 7, 1, 322, 3823, 8092, 3823, 322, 1, 1, 486, 8451, 27876, 23607, 4950, 165, 1, 705, 17205, 82885, 112035, 44803, 4455, 55, 1, 990, 32802, 220198, 440484, 291258
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OFFSET
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0,3
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COMMENTS
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Every fourth row is symmetrical.
Row sums are 4^n.
3*k instead of 4*k in the binomial() gives A178618.
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LINKS
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EXAMPLE
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1;
1, 3;
1, 12, 3;
1, 31, 31, 1;
1, 65, 155, 35;
1, 120, 546, 336, 21;
1, 203, 1554, 1918, 413, 7;
1, 322, 3823, 8092, 3823, 322, 1;
1, 486, 8451, 27876, 23607, 4950, 165;
1, 705, 17205, 82885, 112035, 44803, 4455, 55;
1, 990, 32802, 220198, 440484, 291258, 59950, 2882, 11;
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MAPLE
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(1-x)^(n+1)*add( binomial(n+4*j, 4*j)*x^j, j=0..n+1) ;
coeftayl(%, x=0, k) ;
end proc:
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MATHEMATICA
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p[x_, n_] = (-1)^(n + 1)*(-1 + x)^(n + 1)*Sum[Binomial[n + 4*k, 4*k]*x^k, {k, 0, Infinity}]
Flatten[Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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