A123956 -id:A123956 - cp's OEIS frontend

A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2(-1)^((n-k)/2)(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

1, 0, 1, 2, 0, 1, 0, 1, 0, 1, -2, 0, 0, 0, 1, 0, -3, 0, -1, 0, 1, 2, 0, -3, 0, -2, 0, 1, 0, 5, 0, -2, 0, -3, 0, 1, -2, 0, 8, 0, 0, 0, -4, 0, 1, 0, -7, 0, 10, 0, 3, 0, -5, 0, 1, 2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1, 0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1, -2, 0, 24, 0, -35, 0, 0, 0, 18, 0, -8, 0, 1, 0, -11, 0, 49, 0, -42, 0, -12, 0

Offset: 0

Roger L. Bagula and Gary W. Adamson, Mar 13 2008

Polynomial coefficients of P(n,x) in increasing powers, read by rows, where P(0,x)=1, P(1,x)=x, P(2,x)=2+x^2, P(3,x)=x+x^3, P(4,x)=-2+x^4, and  P(n,x) = x*P(n-1,x) - P(n-2,x) for n>=5.
The row-reversed version of A135929.
Row sums are repeating 1, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1..., see A138034 and A119910.

			{1}, = 1
{0, 1}, = x
{2, 0, 1}, = 2+x^2
{0, 1, 0, 1}, = x+x^3
{-2, 0, 0, 0, 1}, = -2+x^4
{0, -3, 0, -1, 0, 1}, = -3x-x^3+x^5
{2, 0, -3, 0, -2, 0, 1},
{0, 5, 0, -2, 0, -3, 0, 1},
{-2, 0, 8, 0, 0, 0, -4, 0, 1},
{0, -7, 0, 10, 0, 3, 0, -5, 0, 1},
{2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1},
{0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1}

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165

Cf. A123956, A137289.

A137276 := proc(n,k) local nmk,npk; if n = 0 then 1; elif (n-k) mod 2 <> 0 then 0; else nmk := (n-k)/2 ; npk := (n+k)/2 ; (-1)^nmk*(2*k-n)/npk*binomial(npk,nmk) ; fi; end:
seq( seq(A137276(n,k),k=0..n),n=0..13) ;

T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.
P(n,x) = x*P(n-1,x)-P(n-2,x), n>=5.
P(n,2*x) = -2*T(n,x)+4*x*U(n-1,x), where T(n,x) is A053120 and U(n,x) is A053117.

Fourth row inserted by the Associate Editors of the OEIS, Aug 27 2009

A136668 Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2(n-1)P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.

Original entry on oeis.org

1, 0, 1, -2, 0, 2, 0, -11, 0, 8, 8, 0, -74, 0, 48, 0, 119, 0, -632, 0, 384, -48, 0, 1634, 0, -6608, 0, 3840, 0, -1409, 0, 24032, 0, -81984, 0, 46080, 384, 0, -32798, 0, 389312, 0, -1178496, 0, 645120, 0, 18825, 0, -741056, 0, 6966848, 0, -19270656, 0, 10321920

Offset: 1

Roger L. Bagula, Apr 03 2008

Row sums: {1, 1, 0, -3, -18, -129, -1182, -13281, -176478, -2704119, -46909362, ...}.

			Triangle begins as:
    1;
    0,     1;
   -2,     0,    2;
    0,   -11,    0,     8;
    8,     0,  -74,     0,    48;
    0,   119,    0,  -632,     0,    384;
  -48,     0, 1634,     0, -6608,      0, 3840;
    0, -1409,    0, 24032,     0, -81984,    0, 46080;
    ....

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1972, 10th edition, (and various reprintings), p. 631.

G. C. Greubel, Table of n, a(n) for the first 25 rows, flattened

Cf. A106174, A123956.

P[x, 0]= 1; P[x, 1]= 1/x;
P[x_, n_]:= P[x, n] = 2*(n-1)*P[x, n-1]/x - n*P[x, n-2];
Table[ExpandAll[P[x, n] /. x -> 1/y], {n, 0, 10}];
Table[CoefficientList[P[x, n] /. x -> 1/y, y], {n, 0, 10}]//Flatten

P(x,0) = 1; P(x,1) = 1/x; P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2); with substitution of x to 1/y.

cp's OEIS Frontend

A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2(-1)^((n-k)/2)(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

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A136668 Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2(n-1)P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.

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cp's OEIS Frontend

A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

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Maple

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A136668 Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.

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A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2(-1)^((n-k)/2)(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

A136668 Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2(n-1)P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.