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

1,8

Comments

Row sums:

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

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

References

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

Links

Table of n, a(n) for n=1..55.

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}]

Example

{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}

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]

Crossrefs

Cf. A000045, A000930.

Sequence in context: A372585 A200300 A268440 * A318409 A119278 A070064

Adjacent sequences: A137430 A137431 A137432 * A137434 A137435 A137436

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Apr 17 2008

Status

approved