cp's OEIS Frontend

A321203 Irregular triangle T giving the coefficients of x^n = x^{2*e2 + 3*e3} of (1 + x^2 + x^3)^n, with the pair of nonnegative numbers [e2, e3] listed in row n of A321201, for n >= 2.

%I A321203 #13 Dec 23 2018 23:48:51
%S A321203 2,3,6,20,15,20,105,168,70,84,504,1260,252,1320,2310,495,7920,924,
%T A321203 12870,10296,10010,45045,3432,3003,100100,45045,120120,240240,12870,
%U A321203 74256,680680,194480,18564,1113840,1225224,48620,1058148,4232592,831402,542640,8817900,6046560,184756
%N A321203 Irregular triangle T giving the coefficients of x^n = x^{2*e2 + 3*e3} of (1 + x^2  + x^3)^n, with the pair of nonnegative numbers [e2, e3] listed in row n of A321201, for n >= 2.
%C A321203 The row length is r(n), with r(n) = A008615(n+2) for n >= 2:  [1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, ...].
%C A321203 The row sums give A176806(n).
%C A321203 For n = 0 with the trivial [e2, e3] = [0, 0] solution the multinomial is 1 with the row sum A176806(0) = 1. For n = 1 there is no solution (with row sum set to A176806(1) = 0).
%C A321203 This multinomial array for pairs [e2, e3] with 2*2e2 + 3*e3 = n, with nonnegative numbers e2 and e3, is obtained from the multinomial array n!/(e1!*e2!*e3!) with n = e1 + e2 + e3, giving the coefficient x_1^{e1}* x_2^{e2}*x_3^{e3} of (x_1 + x_2 + x_3)^n. Here, in order to find the coefficients of (1 + x^2 + x^3)^n, one sets x_1 = 1, x_2 = x^2 and x_3 = x^3. Hence n = e1 + e2 + e3, and the power of x^n becomes n = 2*e2 + 3*e3. Therefore, e1 = n - (e2 + e3), and the array gives n!/((n-(e2+e3))!*e2!*e3!).
%F A321203 T(n, m) is obtained from the pair(s) [e2, e3] given in row n of A321201 by n!/((n - (e2 +e3))!*e2!*e3!), for n >= 2 and  m = 1, 2, ..., A008615(n+2).
%e A321203 The triangle T(n, m), and the row sums begin:
%e A321203 n\m        0        1        2       3  ...  Row sums A176806(n)
%e A321203 2:         2                                         2
%e A321203 3:         3                                         3
%e A321203 4:         6                                         6
%e A321203 5:        20                                        20
%e A321203 6:        15       20                               35
%e A321203 7:       105                                       105
%e A321203 8:       168       70                              238
%e A321203 9:        84      504                              588
%e A321203 10:     1260      252                             1512
%e A321203 11:     1320     2310                             3630
%e A321203 12:      495     7920      924                    9339
%e A321203 13:    12870    10296                            23166
%e A321203 14:    10010    45045     3432                   58487
%e A321203 15:     3003   100100    45045                  148148
%e A321203 16:   120120   240240    12870                  373230
%e A321203 17:    74256   680680   194480                  949416
%e A321203 18:    18564  1113840  1225224   48620         2406248
%e A321203 19:  1058148  4232592   831402                 6122142
%e A321203 20:   542640  8817900  6046560  184756        15591856
%e A321203 ...
%e A321203 ------------------------------------------------------------------------------
%e A321203 n = 8: (1 + x^2 + x^3)^8 has coefficients 238 of x^n arising from the two [e2, e3] pairs [1, 2] and [4, 0], given in row n = 8 of A321201. The multinomial values are 8!/((8-3)!*1!*2!) = 168 and 8!/((8-4)!*4!*0!) = 70, summing to 238.
%Y A321203 Cf. A008615, A130561, A176806, A321201, A319204.
%K A321203 nonn,tabf
%O A321203 2,1
%A A321203 _Wolfdieter Lang_, Nov 05 2018