A323233 - OEIS

%I #10 Jan 29 2019 04:35:21

%S 1,2,2,6,15,3,24,140,48,4,120,1750,775,110,5,720,28644,14550,2670,210,

%T 6,5040,588588,323008,68775,7105,357,7,40320,14592864,8388800,1962632,

%U 239120,16016,560,8,362880,423227376,250742700,62531532,8502921,680904,32130,828,9

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

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

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

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

%e The triangle starts:

%e [ 1]       1;

%e [ 2]       2,         2;

%e [ 3]       6,        15,         3;

%e [ 4]      24,       140,        48,        4;

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

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

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

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

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

%e The first few polynomials are:

%e p[1](x) = 1;

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

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

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

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

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

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

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

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

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

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

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

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

%K nonn,tabl

%O 1,2

%A _Peter Luschny_, Jan 27 2019