A323233 - OEIS

A323233

Coefficients of polynomials p(n, x) generating the columns of A323224, triangle read by rows, T(n, k) for n >= 1 and k >= 0.

1, 2, 2, 6, 15, 3, 24, 140, 48, 4, 120, 1750, 775, 110, 5, 720, 28644, 14550, 2670, 210, 6, 5040, 588588, 323008, 68775, 7105, 357, 7, 40320, 14592864, 8388800, 1962632, 239120, 16016, 560, 8, 362880, 423227376, 250742700, 62531532, 8502921, 680904, 32130, 828, 9

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..45.

FORMULA

A323224(n, k) = p(k, n)/k!.

T(n, k) = [x^k] p(n, x).

p(n, 1)/n! and p(n, -1)/n! are versions of the partial sums of the Catalan numbers.

EXAMPLE

The triangle starts:

[ 1] 1;

[ 2] 2, 2;

[ 3] 6, 15, 3;

[ 4] 24, 140, 48, 4;

[ 5] 120, 1750, 775, 110, 5;

[ 6] 720, 28644, 14550, 2670, 210, 6;

[ 7] 5040, 588588, 323008, 68775, 7105, 357, 7;

[ 8] 40320, 14592864, 8388800, 1962632, 239120, 16016, 560, 8;

[ 9] 362880, 423227376, 250742700, 62531532, 8502921, 680904, 32130, 828, 9;

The first few polynomials are:

p[1](x) = 1;

p[2](x) = 2*x + 2!;

p[3](x) = 3*x*(x + 5) + 3!;

p[4](x) = 4*x*(x + 5)*(x + 7) + 4!;

p[5](x) = 5*x*(x + 5)*(x + 7)*(x + 10) + 5!;

p[6](x) = 6*x*(x + 7)*(x + 11)*(x^2 + 17*x + 62) + 6!;

p[7](x) = 7*x*(x + 6)*(x + 7)*(x + 11)*(x + 13)*(x + 14) + 7!;

MATHEMATICA

ogf[n_] := (2/(1 + Sqrt[1 - 4 x] ))^n x/(1 - x);

ser[n_, len_] := CoefficientList[Series[ogf[n], {x, 0, (n + 1) len + 1}], x];

tab[k_, len_] := Table[{n, ser[n, k + 1][[k + 1]]}, {n, 0, len - 1}];

pol[n_] := n! InterpolatingPolynomial[tab[n, n + 1], x] // Expand;

row[n_] := CoefficientList[pol[n], x]; Table[row[n], {n, 1, 9}]

CROSSREFS

Cf. A323224, A034856, A323221, A323220, A014137, A014138.

Sequence in context: A201687 A259810 A142471 * A071208 A231536 A216242

Adjacent sequences: A323230 A323231 A323232 * A323234 A323235 A323236

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jan 27 2019

STATUS

approved