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.

2, 3, 6, 20, 15, 20, 105, 168, 70, 84, 504, 1260, 252, 1320, 2310, 495, 7920, 924, 12870, 10296, 10010, 45045, 3432, 3003, 100100, 45045, 120120, 240240, 12870, 74256, 680680, 194480, 18564, 1113840, 1225224, 48620, 1058148, 4232592, 831402, 542640, 8817900, 6046560, 184756

Offset: 2

Author

Wolfdieter Lang, Nov 05 2018

Keywords

• nonn
• tabf

Comments

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, ...].

The row sums give A176806(n).

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).

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!).

Examples

			The triangle T(n, m), and the row sums begin:
n\m        0        1        2       3  ...  Row sums A176806(n)
2:         2                                         2
3:         3                                         3
4:         6                                         6
5:        20                                        20
6:        15       20                               35
7:       105                                       105
8:       168       70                              238
9:        84      504                              588
10:     1260      252                             1512
11:     1320     2310                             3630
12:      495     7920      924                    9339
13:    12870    10296                            23166
14:    10010    45045     3432                   58487
15:     3003   100100    45045                  148148
16:   120120   240240    12870                  373230
17:    74256   680680   194480                  949416
18:    18564  1113840  1225224   48620         2406248
19:  1058148  4232592   831402                 6122142
20:   542640  8817900  6046560  184756        15591856
...
------------------------------------------------------------------------------
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.
		

Crossrefs

Cf. A008615, A130561, A176806, A321201, A319204.

Formula

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).