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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. 6
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS 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

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Last modified January 22 04:27 EST 2020. Contains 331133 sequences. (Running on oeis4.)