A049323 Triangle of coefficients of certain polynomials (exponents in increasing order), equivalent to A033842.

1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 50, 125, 125, 1, 15, 120, 540, 1296, 1296, 1, 21, 245, 1715, 7203, 16807, 16807, 1, 28, 448, 4480, 28672, 114688, 262144, 262144, 1, 36, 756, 10206, 91854, 551124, 2125764, 4782969, 4782969, 1, 45, 1200, 21000, 252000

Offset

0,5

Comments

These polynomials p(n, x) appear in the W. Lang reference as c1(-(n+1);x), n >= 0 on p.12. The coefficients are given there in eq.(44) on p. 6. - Wolfdieter Lang, Nov 20 2015

Links

Table of n, a(n) for n=0..49.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Formula

a(n, m) = A033842(n, n-m) = binomial(n+1, m+1)*(n+1)^{m-1}, n >= m >= 0, else 0.

p(k-1, -x)/(1-k*x)^k =(-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.

From Werner Schulte, Oct 19 2015: (Start)

a(2*n,n) = A000108(n)*(2*n+1)^n;

a(3*n,2*n) = A001764(n)*(3*n+1)^(2*n);

a(p*n,(p-1)*n) = binomial(p*n,n)/((p-1)*n+1)*(p*n+1)^((p-1)*n) for p > 0;

Sum_{m=0..n} (m+1)*a(n,m) = (n+2)^n;

Sum_{m=0..n} (-1)^m*(m+1)*a(n,m) = (-n)^n where 0^0 = 1;

p(n,x) = Sum_{m=0..n} a(n,m)*x^m = ((1+(n+1)*x)^(n+1)-1)/((n+1)^2*x).

(End)

Example

The triangle a(n, m) begins:
n\m 0  1   2    3     4      5      6      7 ...
0:  1
1:  1  1
2:  1  3   3
3:  1  6  16   16
4:  1 10  50  125  125
5:  1 15 120  540  1296  1296
6:  1 21 245 1715  7203  16807  16807
7:  1 28 448 4480 28672 114688 262144 262144
... reformatted. - Wolfdieter Lang, Nov 20 2015
E.g. the third row {1,3,3} corresponds to polynomial p(2,x)= 1 + 3*x + 3*x^2.

Maple

seq(seq(binomial(n+1, m+1)*(n+1)^(m-1), m=0..n), n=0..10); # Robert Israel, Oct 19 2015

Mathematica

Table[Binomial[n + 1, k + 1] (n + 1)^(k - 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 19 2015 *)

Prog

(Magma) /* As triangle: */ [[Binomial(n+1, k+1)*(n+1)^(k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 20 2015

Crossrefs

a(n, 0)= A000012 (powers of 1), a(n, 1)= A000217 (triangular numbers), a(n, n)= A000272(n+1), n >= 0 (diagonal), a(n, n-1)= A000272(n+1), n >= 1.

For n = 0..5 the row sequences a(n, m), m >= 0, are the first columns of the triangles A023531 (unit matrix), A030528, A049324, A049325, A049326, A049327, respectively.

Cf. A033842, A046757.

Sequence in context: A138464 A117279 A234251 * A322148 A084144 A306759

Adjacent sequences: A049320 A049321 A049322 * A049324 A049325 A049326

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang

Status

approved