A137433 Coefficients of A000930 expansion similar to that given for Fibonacci numbers in Roman's Umbral Calculus.

1, 0, 1, 0, 1, 1, 0, 8, 3, 1, 0, 30, 35, 6, 1, 0, 144, 230, 95, 10, 1, 0, 1200, 1954, 945, 205, 15, 1, 0, 10800, 19824, 11494, 2835, 385, 21, 1, 0, 105840, 216012, 149212, 45409, 7000, 658, 28, 1, 0, 1249920, 2692080, 2055500, 740124, 140889, 15120, 1050, 36, 1

Offset: 1

Author

Roger L. Bagula, Apr 17 2008

Keywords

• nonn
• tabl
• uned

Comments

Row sums:

{1, 1, 2, 12, 72, 480, 4320, 45360, 524160, 6894720, 101606400}

Row_sum(n)/n!=A000930(n)

Examples

			{1},
{0, 1},
{0, 1, 1},
{0, 8, 3, 1},
{0, 30, 35, 6, 1},
{0, 144, 230, 95, 10, 1},
{0, 1200, 1954, 945, 205, 15, 1},
{0, 10800, 19824, 11494, 2835, 385, 21, 1},
{0, 105840, 216012, 149212, 45409, 7000, 658, 28, 1},
{0, 1249920, 2692080, 2055500, 740124, 140889, 15120, 1050, 36, 1},
{0, 16692480, 37802736, 31266540, 12628160, 2814525, 370713, 29610, 1590, 45, 1}
		

References

• Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150

Crossrefs

Cf. A000045, A000930.

Programs

Mathematica

Clear[p, g]; p[t_] = 1/(1 - t - t^3)^x; Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]

Formula

Coefficients expansion of p(x,n) in f(x,t)=1/(1-t-t^3)^x=Sum[p(x,n)*t^n/n!m{n,1,Infinity}]