This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309951 #89 Jul 14 2025 15:20:54
%S A309951 1,1,1,1,1,3,2,1,10,27,18,1,47,718,4416,10656,6912,1,246,20545,751800,
%T A309951 12911500,100380000,304200000,216000000,1,1602,929171,260888070,
%U A309951 39883405500,3492052425000,177328940580000,5153150631600000,82577533320000000,669410956800000000,2224399449600000000,1632586752000000000,1,11481
%N A309951 Irregular triangular array, read by rows: T(n,k) is the sum of the products of multinomial coefficients (n_1 + n_2 + n_3 + ...)!/(n_1! * n_2! * n_3! * ...) taken k at a time, where (n_1, n_2, n_3, ...) runs over all integer partitions of n (n >= 0, 0 <= k <= A000041(n)).
%C A309951 This array was inspired by _R. H. Hardin_'s recurrences for the columns of array A212855. Rows k=1 to k=5 are due to him, while the remaining rows were computed by _Alois P. Heinz_.
%C A309951 Row n has length A000041(n) + 1, i.e., one more than the number of partitions of n.
%C A309951 Let R(m,n) := R(m,n,t=0) = A212855(m,n) for m,n >= 1, where R(m,n,t) = LHS of Eq. (6) of Abramson and Promislow (1978, p. 248).
%C A309951 Let P_n be the set of all lists a = (a_1, a_2,..., a_n) of integers a_i >= 0, i = 1,..., n such that 1*a_1 + 2*a_2 + ... + n*a_n = n; i.e., P_n is the set all integer partitions of n. (We use a different notation for partitions than the one in the name of T(n,k).) Then |P_n| = A000041(n) for n >= 0.
%C A309951 We have R(m,n) = A212855(m,n) = Sum_{a in P_n} (-1)^(n - Sum_{j=1..n} a_j) * (a_1 + a_2 + ... + a_n)!/(a_1! * a_2! * ... * a_n!) * (n! / ((1!)^a_1 * (2!)^a_2 * ... * (n!)^a_n))^m.
%C A309951 The recurrence of _R. H. Hardin_ for column n of array A212855 is Sum_{s = 0..|P_n|} (-1)^s * T(n,s) * R(m-s,n) = 0 for n >= 1 and m >= |P_n| + 1.
%C A309951 The above recurrence is correct for all n >= 1, but it is not always a minimal one. For example, it seems to be the minimal one for n = 1,...,6, but not for n = 7 (see A212854). It seems to be minimal whenever every two different partitions of n give different multinomial coefficients.
%C A309951 For n = 7, the partitions (a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (0, 2, 1, 0, 0, 0, 0) (i.e., 2 + 2 + 3) and (a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (3, 0, 0, 1, 0, 0, 0) (i.e., 1 + 1 + 1 + 4) give the same multinomial coefficient: 210 = 7!/(2!2!3!) = 7!/(1!1!1!4!). Hence, to find the minimal recurrence for n = 7, we count 210 only once in the set of multinomial coefficients: 1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040. Then the absolute value of the coefficient of a(n-1) in the minimal recurrence is the sum of these multinomial coefficients (i.e., 11271); the absolute value of the coefficient of a(n-2) in the minimal recurrence is the sum of products of every two of them (i.e., 46169368), and so on.
%C A309951 Looking at the multinomial coefficients of the integer partitions of n = 8, 9, 10 on pp. 831-832 of Abramowitz and Stegun (1964), we see that, even in these cases, the above recurrence is not the minimal one. The number of distinct multinomial coefficients among the integer partitions of n is given by A070289.
%H A309951 Alois P. Heinz, <a href="/A309951/b309951.txt">Rows n = 0..14, flattened</a>
%H A309951 Milton Abramowitz and Irene A. Stegun, <a href="https://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>, National Bureau of Standards (Applied Mathematics Series, 55), 1964; see pp. 831-832 for the multinomial coefficients of integer partitions of n = 1..10.
%H A309951 Morton Abramson and David Promislow, <a href="https://doi.org/10.1016/0097-3165(78)90012-2">Enumeration of arrays by column rises</a>, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (6), p. 248, and the comments above.
%H A309951 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>.
%F A309951 Sum_{k=0..A000041(n)} (-1)^k * T(n,k) = 0.
%e A309951 Triangle begins as follows:
%e A309951 [n=0]: 1, 1;
%e A309951 [n=1]: 1, 1;
%e A309951 [n=2]: 1, 3, 2;
%e A309951 [n=3]: 1, 10, 27, 18;
%e A309951 [n=4]: 1, 47, 718, 4416, 10656, 6912;
%e A309951 [n=5]: 1, 246, 20545, 751800, 12911500, 100380000, 304200000, 216000000;
%e A309951 ...
%e A309951 For example, when n = 3, the integer partitions of 3 are 3, 1+2, 1+1+1, and the corresponding multinomial coefficients are 3!/3! = 1, 3!/(1!2!) = 3, and 3!/(1!1!1!) = 6. Then T(n=3, k=0) = 1, T(n=3, k=1) = 1 + 3 + 6 = 10, T(n=3, k=2) = 1*3 + 1*6 + 3*6 = 27, and T(n=3, k=3) = 1*3*6 = 18.
%e A309951 Since |P_3| = A000041(3) = 3, the recurrence of _R. H. Hardin_ for column n = 3 of array A212855 is T(3,0)*R(m,3) - T(3,1)*R(m-1,3) + T(3,2)*R(m-2,3) - T(3,3)*R(m-3,3) = 0; i.e., R(m,3) - 10*R(m-1,3) + 27*R(m-2,3) - 18*R(m-3,3) = 0 for m >= 4. We have the initial conditions R(m=1,3) = 1, R(m=2,3) = 19, and R(m=3,3) = 163. Thus, R(m,3) = 6^m - 2*3^m + 1 = A212850(m) for m >= 1. See the documentation of array A212855.
%p A309951 g:= proc(n, i) option remember; `if`(n=0 or i=1, [n!], [map(x->
%p A309951 binomial(n, i)*x, g(n-i, min(n-i, i)))[], g(n, i-1)[]])
%p A309951 end:
%p A309951 b:= proc(n, m) option remember; `if`(n=0, 1,
%p A309951 expand(b(n-1, m)*(g(m$2)[n]*x+1)))
%p A309951 end:
%p A309951 T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(nops(g(n$2)), n)):
%p A309951 seq(T(n), n=0..7); # _Alois P. Heinz_, Aug 25 2019
%t A309951 g[n_, i_] := g[n, i] = If[n==0 || i==1, {n!}, Join[Binomial[n, i]*#& /@ g[n - i, Min[n - i, i]], g[n, i - 1]]];
%t A309951 b[n_, m_] := b[n, m] = If[n==0, 1, Expand[b[n-1, m]*(g[m, m][[n]]*x+1)]];
%t A309951 T[n_] := CoefficientList[b[Length[g[n, n]], n], x];
%t A309951 T /@ Range[0, 7] // Flatten (* _Jean-François Alcover_, Feb 18 2021, after _Alois P. Heinz_ *)
%Y A309951 Cf. A000012 (column k=0), A000041, A005651 (column k=1), A070289, A212850, A212851, A212852, A212853, A212854, A212855, A212856, A212857, A212858, A212859, A212860.
%Y A309951 Rightmost terms in rows give A309972.
%K A309951 nonn,tabf
%O A309951 0,6
%A A309951 _Petros Hadjicostas_, Aug 25 2019