A183610 Rectangular table where T(n,k) is the sum of the n-th powers of the k-th row of multinomial coefficients in triangle A036038 for n>=0, k>=0, as read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 5, 1, 1, 9, 46, 47, 7, 1, 1, 17, 244, 773, 246, 11, 1, 1, 33, 1378, 15833, 19426, 1602, 15, 1, 1, 65, 8020, 354065, 1980126, 708062, 11481, 22, 1, 1, 129, 47386, 8220257, 221300626, 428447592, 34740805, 95503, 30

Offset

0,6

Links

Paul D. Hanna, Table of rows 0..32 as read by antidiagonals, with index n = 0..528.

Formula

G.f. of row n: Sum_{k>=0} T(n,k)*x^k/k!^n = Product_{j>=1} 1/(1 - x^j/j!^n).

Example

The table begins:
n=0: [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...];
n=1: [1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, ...];
n=2: [1, 1, 5, 46, 773, 19426, 708062, 34740805, 2230260741, ...];
n=3: [1, 1, 9, 244, 15833, 1980126, 428447592, 146966837193, ...];
n=4: [1, 1, 17, 1378, 354065, 221300626, 286871431922, ...];
n=5: [1, 1, 33, 8020, 8220257, 25688403126, 199758931567152, ...];
n=6: [1, 1, 65, 47386, 194139713, 3033434015626, 141528428949437282, ...];
n=7: [1, 1, 129, 282124, 4622599553, 361140600078126, ...];
n=8: [1, 1, 257, 1686178, 110507041025, 43166813000390626, ...];
n=9: [1, 1, 513, 10097380, 2646977660417, 5169878244001953126, ...];
n=10:[1, 1, 1025, 60525226, 63465359844353, 619778904740009765626, ...];
...
The sums of the n-th power of terms in row k of triangle A036038 begin:
T(n,1) = 1^n,
T(n,2) = 1^n + 2^n,
T(n,3) = 1^n + 3^n + 6^n,
T(n,4) = 1^n + 4^n + 6^n + 12^n + 24^n,
T(n,5) = 1^n + 5^n + 10^n + 20^n + 30^n + 60^n + 120^n,
T(n,6) = 1^n + 6^n + 15^n + 20^n + 30^n + 60^n + 90^n + 120^n + 180^n + 360^n + 720^n, ...
Note that row n=0 forms the partition numbers A000041.

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i), k)/i!^k+b(n, i-1, k))
    end:
A:= (n, k)-> k!^n*b(k$2, n):
seq(seq(A(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Sep 11 2019

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, 1, b[n-i, Min[n-i, i], k]/i!^k + b[n, i-1, k]];
A[n_, k_] := k!^n b[k, k, n];
Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

Prog

(PARI) {T(n, k)=k!^n*polcoeff(1/prod(m=1, k, 1-x^m/m!^n +x*O(x^k)), k)}
for(n=0, 10, for(k=0, 8, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A036038, A000041, A005651, A183240, A183235, A183236, A183237, A183238, A215910 (main diagonal).

Sequence in context: A128545 A194672 A034364 * A261365 A261507 A304942

Adjacent sequences: A183607 A183608 A183609 * A183611 A183612 A183613

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 11 2012

Status

approved