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A134090
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Triangle, read by rows, where T(n,k) = [(I + D*C)^n](n,k); that is, row n of T = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere.
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5
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1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 71, 46, 18, 4, 1, 456, 285, 110, 30, 5, 1, 3337, 2021, 780, 215, 45, 6, 1, 27203, 16023, 6167, 1729, 371, 63, 7, 1, 243203, 139812, 53494, 15176, 3346, 588, 84, 8, 1, 2357356, 1326111, 504030, 143814, 32376, 5886, 876, 108, 9, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Column 0 equals A122455 if we define A122455(0)=1.
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FORMULA
| T(n,k) = [x^(n-k)] Sum_{j=0..n} C(n,j)*x^j/(1-j*x)^k /[Product_{i=0..j}(1-i*x)].
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EXAMPLE
| Triangle T begins:
1;
1, 1;
3, 2, 1;
13, 9, 3, 1;
71, 46, 18, 4, 1;
456, 285, 110, 30, 5, 1;
3337, 2021, 780, 215, 45, 6, 1;
27203, 16023, 6167, 1729, 371, 63, 7, 1;
243203, 139812, 53494, 15176, 3346, 588, 84, 8, 1;
2357356, 1326111, 504030, 143814, 32376, 5886, 876, 108, 9, 1; ...
Let P denote the matrix equal to Pascal's triangle shift down 1 row:
P(n,k) = C(n+1,k) for n>k>=0, with P(n,n)=1 for n>=0.
Illustrate row n of T = row n of P^n as follows.
Matrix P = I + D*C begins:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 3, 3, 1, 1;
1, 4, 6, 4, 1, 1; ...
Matrix cube P^3 begins:
1;
3, 1;
6, 3, 1;
13, 9, 3, 1; <== row 3 of P^3 = row 3 of T
30, 25, 12, 3, 1;
73, 72, 40, 15, 3, 1; ...
Matrix 4-th power P^4 begins:
1;
4, 1;
10, 4, 1;
26, 14, 4, 1;
71, 46, 18, 4, 1; <== row 4 of P^4 = row 4 of T
204, 155, 70, 22, 4, 1; ...
Matrix 5-th power P^5 begins:
1;
5, 1;
15, 5, 1;
45, 20, 5, 1;
140, 75, 25, 5, 1;
456, 285, 110, 30, 5, 1; <== row 5 of P^5 = row 5 of T.
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PROG
| (PARI) /* As generated by the g.f.: */ {T(n, k)=polcoeff(sum(j=0, n, binomial(n, j)*x^j/(1-j*x)^k/prod(i=0, j, 1-i*x+x*O(x^(n-k)))), n-k)} /* As generated by matrix power: row n of T equals row n of P^n: */ {T(n, k)=local(P=matrix(n+1, n+1, r, c, if(r==c, 1, if(r>c, binomial(r-2, c-1))))); (P^n)[n+1, k+1]}
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CROSSREFS
| Cf. columns:, A134091, A134092, A134093; A134094 (row sums).
Sequence in context: A059438 A156628 A104980 * A132845 A129652 A154921
Adjacent sequences: A134087 A134088 A134089 * A134091 A134092 A134093
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Oct 07 2007
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