|
|
A156136
|
|
A triangle of polynomial coefficients related to Mittag-Leffler polynomials: p(x,n)=Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x).
|
|
1
|
|
|
1, 2, 2, 3, 12, 4, 4, 36, 48, 8, 5, 80, 240, 160, 16, 6, 150, 800, 1200, 480, 32, 7, 252, 2100, 5600, 5040, 1344, 64, 8, 392, 4704, 19600, 31360, 18816, 3584, 128, 9, 576, 9408, 56448, 141120, 150528, 64512, 9216, 256, 10, 810, 17280, 141120, 508032
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
REFERENCES
|
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 75-76
|
|
LINKS
|
|
|
FORMULA
|
p(x,n)=Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x);
p(x,n)=n Hypergeometric2F1[1 - n, 1 - n, 2, 2 x];
t(n,m)=coefficiemts(p(x,n))
|
|
EXAMPLE
|
1;
2, 2;
3, 12, 4;
4, 36, 48, 8;
5, 80, 240, 160, 16;
6, 150, 800, 1200, 480, 32;
7, 252, 2100, 5600, 5040, 1344, 64;
8, 392, 4704, 19600, 31360, 18816, 3584, 128;
9, 576, 9408, 56448, 141120, 150528, 64512, 9216, 256;
10, 810, 17280, 141120, 508032, 846720, 645120, 207360, 23040, 512;
|
|
MATHEMATICA
|
Clear[t0, p, x, n, m];
p[x_, n_] = Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|