A253383 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x-3k)^k.
1, 7, 2, 7, 38, 3, 7, 362, 111, 4, 7, 2522, 2271, 244, 5, 7, 14672, 34671, 8344, 455, 6, 7, 75908, 442911, 212464, 23135, 762, 7, 7, 361676, 5015199, 4498984, 869855, 53682, 1183, 8, 7, 1621388, 52044447, 83860840, 26997215, 2775282, 110047, 1736, 9, 7, 6935798, 505540767, 1423092160, 732435935, 117592782, 7458367, 205856, 2439, 10
Offset: 0
Examples
The triangle T(n,k) starts: n\k 0 1 2 3 4 5 6 7 8 ... 0: 1 1: 7 2 2: 7 38 3 3: 7 362 111 4 4: 7 2522 2271 244 5 5: 7 14672 34671 8344 455 6 6: 7 75908 442911 212464 23135 762 7 7: 7 361676 5015199 4498984 869855 53682 1183 8 8: 7 1621388 52044447 83860840 26997215 2775282 110047 1736 9 ... ----------------------------------------------------------------- n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 7*(x-0)^0 + 362*(x-3)^1 + 111*(x-6)^2 + 4*(x-9)^3.
Programs
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PARI
T(n, k)=(k+1)-sum(i=k+1, n, (-3*i)^(i-k)*binomial(i, k)*T(n, i)) for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
Formula
T(n,n-1) = n + 3*n^2 + 3*n^3, for n >= 1.
T(n,n-2) = (n-1)*(9*n^4 - 9*n^3 - 12*n^2 - 6*n + 2)/2, for n >= 2.
T(n,n-3) = (n-2)*(9*n^6 - 54*n^5 + 81*n^4 + 9*n^3 - 12*n^2 - 45*n + 14)/2, for n >= 3.
Comments