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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.

%I A183610 #19 Dec 06 2019 09:58:48
%S A183610 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,
%T A183610 1,33,1378,15833,19426,1602,15,1,1,65,8020,354065,1980126,708062,
%U A183610 11481,22,1,1,129,47386,8220257,221300626,428447592,34740805,95503,30
%N 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.
%H A183610 Paul D. Hanna, <a href="/A183610/b183610.txt">Table of rows 0..32 as read by antidiagonals, with index n = 0..528.</a>
%F A183610 G.f. of row n: Sum_{k>=0} T(n,k)*x^k/k!^n = Product_{j>=1} 1/(1 - x^j/j!^n).
%e A183610 The table begins:
%e A183610 n=0: [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...];
%e A183610 n=1: [1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, ...];
%e A183610 n=2: [1, 1, 5, 46, 773, 19426, 708062, 34740805, 2230260741, ...];
%e A183610 n=3: [1, 1, 9, 244, 15833, 1980126, 428447592, 146966837193, ...];
%e A183610 n=4: [1, 1, 17, 1378, 354065, 221300626, 286871431922, ...];
%e A183610 n=5: [1, 1, 33, 8020, 8220257, 25688403126, 199758931567152, ...];
%e A183610 n=6: [1, 1, 65, 47386, 194139713, 3033434015626, 141528428949437282, ...];
%e A183610 n=7: [1, 1, 129, 282124, 4622599553, 361140600078126, ...];
%e A183610 n=8: [1, 1, 257, 1686178, 110507041025, 43166813000390626, ...];
%e A183610 n=9: [1, 1, 513, 10097380, 2646977660417, 5169878244001953126, ...];
%e A183610 n=10:[1, 1, 1025, 60525226, 63465359844353, 619778904740009765626, ...];
%e A183610 ...
%e A183610 The sums of the n-th power of terms in row k of triangle A036038 begin:
%e A183610 T(n,1) = 1^n,
%e A183610 T(n,2) = 1^n + 2^n,
%e A183610 T(n,3) = 1^n + 3^n + 6^n,
%e A183610 T(n,4) = 1^n + 4^n + 6^n + 12^n + 24^n,
%e A183610 T(n,5) = 1^n + 5^n + 10^n + 20^n + 30^n + 60^n + 120^n,
%e A183610 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, ...
%e A183610 Note that row n=0 forms the partition numbers A000041.
%p A183610 b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
%p A183610       b(n-i, min(n-i, i), k)/i!^k+b(n, i-1, k))
%p A183610     end:
%p A183610 A:= (n, k)-> k!^n*b(k$2, n):
%p A183610 seq(seq(A(d-k, k), k=0..d), d=0..10);  # _Alois P. Heinz_, Sep 11 2019
%t A183610 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]];
%t A183610 A[n_, k_] := k!^n b[k, k, n];
%t A183610 Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, after _Alois P. Heinz_ *)
%o A183610 (PARI) {T(n,k)=k!^n*polcoeff(1/prod(m=1, k, 1-x^m/m!^n +x*O(x^k)), k)}
%o A183610 for(n=0,10,for(k=0,8,print1(T(n,k),", "));print(""))
%Y A183610 Cf. A036038, A000041, A005651, A183240, A183235, A183236, A183237, A183238, A215910 (main diagonal).
%K A183610 nonn,tabl
%O A183610 0,6
%A A183610 _Paul D. Hanna_, Aug 11 2012