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A321203
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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.
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4
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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
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OFFSET
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2,1
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COMMENTS
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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, ...].
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!).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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
...
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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.
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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