A039757 -id:A039757 - cp's OEIS frontend

A109409 Coefficients of polynomials triangular sequence produced by removing primes from the odd numbers in A028338.

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 9, 10, 1, 0, 0, 0, 0, 9, 10, 1, 0, 0, 0, 0, 0, 9, 10, 1, 0, 0, 0, 0, 0, 135, 159, 25, 1, 0, 0, 0, 0, 0, 0, 135, 159, 25, 1, 0, 0, 0, 0, 0, 0, 0, 135, 159, 25, 1, 0, 0, 0, 0, 0, 0, 0, 2835, 3474, 684, 46, 1

Offset: 1

Roger L. Bagula, May 19 2007

The row sums also appear to be new: b = Flatten[Join[{{1}}, Table[Apply[Plus, Abs[CoefficientList[Product[x + g[n], {n, 0, m}], x]]], {m, 0, 10}]]] {1, 2, 2, 2, 2, 20, 20, 20, 320, 320, 320, 7040} Since the row sum of A028338 is the double factorial A000165: this result seems to be a factorization of the double factorial numbers by relatively sparse nonprime odd numbers. It might be better to reverse the order of the coefficients to get the higher powers first.

			{1},
{1, 1},
{0, 1, 1},
{0, 0, 1, 1},
{0, 0, 0, 1, 1},
{0, 0, 0, 9, 10, 1},
{0, 0, 0, 0, 9, 10, 1},
{0, 0, 0, 0, 0, 9, 10, 1}

Cf. A028338, A000165, A039757.

a = Join[{{1}}, Table[CoefficientList[Product[x + g[n], {n, 0, m}], x], {m, 0, 10}]]; Flatten[a]

p[n]=Product[If[PrimeQ[2*n+1]==false,x+(2*n+1),x] a(n) =CoefficientList[p[n],x]

A155718 Symmetrical form of A039683 using polynomials: p(x,n)=Product[x - (2i), {i, 0, Floor[n/2]}]/x; t(n,m)=coefficients(p(x,n)+x^np(1/x,n)); t(n,m)=A039683(n,m)+A039683(n,n-m).

Original entry on oeis.org

2, -1, -1, 9, -12, 9, -47, 32, 32, -47, 385, -420, 280, -420, 385, -3839, 4354, -1460, -1460, 4354, -3839, 46081, -56490, 26684, -11760, 26684, -56490, 46081, -645119, 836296, -418936, 92624, 92624, -418936, 836296, -645119, 10321921, -14026824

Offset: 0

Roger L. Bagula, Jan 25 2009

Row sums are:
{2, -2, 6, -30, 210, -1890, 20790, -270270, 4054050, -68918850, 1309458150,...}.
The Stirling product form is: as even- odd factorization;
Product[x-i,{i,0,n}]=Product[x-(2*i),{i,0,Floor[n/2]}]*Product[x-(2*i+1),{i,0,Floor[n/2]}]

			{2},
{-1, -1},
{9, -12, 9},
{-47, 32, 32, -47},
{385, -420, 280, -420, 385},
{-3839, 4354, -1460, -1460, 4354, -3839},
{46081, -56490, 26684, -11760, 26684, -56490, 46081},
{-645119, 836296, -418936, 92624, 92624, -418936, 836296, -645119},
{10321921, -14026824, 7562120, -2189376, 718368, -2189376, 7562120, -14026824, 10321921},
{-185794559, 262803366, -150102120, 46239920, -7606032, -7606032, 46239920, -150102120, 262803366, -185794559},
{3715891201, -5441863790, 3264920736, -1076561200, 221207888, -57731520, 221207888, -1076561200, 3264920736, -5441863790, 3715891201}

A039683, A039757

Clear[p, x, n, b, a, b0];
p[x_, n_] := Product[x - (2*i), {i, 0, Floor[n/2]}]/x;
Table[Expand[ CoefficientList[ExpandAll[p[x, n]], x] + Reverse[CoefficientList[ExpandAll[p[x, n]], x]]], {n, 0, 20, 2}];
Flatten[%]

p(x,n)=Product[x - (2*i), {i, 0, Floor[n/2]}]/x;
t(n,m)=coefficients(p(x,n)+x^n*p(1/x,n));
t(n,m)=A039683(n,m)+A039683(n,n-m).

cp's OEIS Frontend

A109409 Coefficients of polynomials triangular sequence produced by removing primes from the odd numbers in A028338.

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A155718 Symmetrical form of A039683 using polynomials: p(x,n)=Product[x - (2i), {i, 0, Floor[n/2]}]/x; t(n,m)=coefficients(p(x,n)+x^np(1/x,n)); t(n,m)=A039683(n,m)+A039683(n,n-m).

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cp's OEIS Frontend

A109409 Coefficients of polynomials triangular sequence produced by removing primes from the odd numbers in A028338.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A155718 Symmetrical form of A039683 using polynomials: p(x,n)=Product[x - (2*i), {i, 0, Floor[n/2]}]/x; t(n,m)=coefficients(p(x,n)+x^n*p(1/x,n)); t(n,m)=A039683(n,m)+A039683(n,n-m).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A155718 Symmetrical form of A039683 using polynomials: p(x,n)=Product[x - (2i), {i, 0, Floor[n/2]}]/x; t(n,m)=coefficients(p(x,n)+x^np(1/x,n)); t(n,m)=A039683(n,m)+A039683(n,n-m).