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URL: https://oeis.org/A146769

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A146769
Coefficients of polynomial P(n) by rows, with P(n) = (x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1) for n > 0 and P(0) = 1.
1
1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 12, 18, 12, 1, 1, 25, 50, 50, 25, 1, 1, 54, 135, 180, 135, 54, 1, 1, 119, 357, 595, 595, 357, 119, 1, 1, 264, 924, 1848, 2310, 1848, 924, 264, 1, 1, 585, 2340, 5460, 8190, 8190, 5460, 2340, 585, 1, 1, 1290, 5805, 15480, 27090, 32508
OFFSET
0,5
COMMENTS
Original name: A new symmetrical polynomial form to give a triangle sequence: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
Row sums are:{1, 2, 5, 14, 44, 152, 560, 2144, 8384, 33152, 131840}.
Row sums are 1 and (6*2^k + 4^k)/8 for k >= 1 (see A257273). - Robert Israel, Apr 29 2015
FORMULA
G.f.: y/(4*(2*y-1)) - 1/(x*y+y-1) - 1/(8*(2*x*y+2*y-1)) + 1/(8*(2*x*y-1)). - Robert Israel, Apr 29 2015
EXAMPLE
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 12, 18, 12, 1;
1, 25, 50, 50, 25, 1;
1, 54, 135, 180, 135, 54, 1;
1, 119, 357, 595, 595, 357, 119, 1;
1, 264, 924, 1848, 2310, 1848, 924, 264, 1;
1, 585, 2340, 5460, 8190, 8190, 5460, 2340, 585, 1;
1, 1290, 5805, 15480, 27090, 32508, 27090, 15480, 5805, 1290, 1;
...
MATHEMATICA
p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]
PROG
(Magma) /* As triangle: */ [1]; for n in [1..10] do; R<x> := PolynomialAlgebra(RationalField(), n); Coefficients((x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1)); end for; // Bruno Berselli, Apr 30 2015
CROSSREFS
Cf. A257273 (row sums).
Sequence in context: A131235 A202812 A157243 * A189610 A172427 A143362
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Nov 02 2008
EXTENSIONS
New name from Charles R Greathouse IV, Apr 29 2015
STATUS
approved