A355259 Triangle read by rows. Row k are the coefficients of the polynomials (sorted by ascending powers) which interpolate the points (n, A355257(n, k+1)) for n = 0..k.

1, 3, 2, 14, 12, 3, 90, 82, 30, 4, 744, 680, 285, 60, 5, 7560, 6788, 2985, 760, 105, 6, 91440, 80136, 35532, 9870, 1715, 168, 7, 1285200, 1098984, 482300, 138796, 27160, 3444, 252, 8, 20603520, 17227584, 7425492, 2152584, 447405, 65520, 6342, 360, 9

Offset: 0

Author

Peter Luschny, Jul 03 2022

Keywords

• nonn
• tabl

Comments

Conjecture from Mélika Tebni: These polynomials generate column k + 1 of

A355257.

Examples

			[0]        1;
[1]        3,        2;
[2]       14,       12,       3;
[3]       90,       82,      30,       4;
[4]      744,      680,     285,      60,      5;
[5]     7560,     6788,    2985,     760,    105,     6;
[6]    91440,    80136,   35532,    9870,   1715,   168,    7;
[7]  1285200,  1098984,  482300,  138796,  27160,  3444,  252,   8;
[8] 20603520, 17227584, 7425492, 2152584, 447405, 65520, 6342, 360, 9;
.
Seen as polynomials:
p0(x) = 1;
p1(x) = 3 + 2*x;
p2(x) = 14 + 12*x + 3*x^2;
p3(x) = 90 + 82*x + 30*x^2 + 4*x^3;
p4(x) = 744 + 680*x + 285*x^2 + 60*x^3 + 5*x^4;
p5(x) = 7560 + 6788*x + 2985*x^2 + 760*x^3 + 105*x^4 + 6*x^5;
p6(x) = 91440 + 80136*x + 35532*x^2 + 9870*x^3 + 1715*x^4 + 168*x^5 + 7*x^6;
		

Crossrefs

Cf. A355257.

Programs

Maple

A355257 := (n, k) -> add(k!*binomial(k + n - 1, k - j - 1)/(j + 1), j = 0..k-1):
for k from 0 to 9 do CurveFitting:-PolynomialInterpolation([seq([n, A355257(n, k+1)], n = 0..k)], x):
print(seq(coeff(%, x, j), j = 0..k)) od: