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A324960
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Triangle of coefficients T(n,k) of y^k in Product_{k=0..n-1} (1 + (k+2)*y + y^2), read by rows of terms k = 0..2*n, for n >= 0.
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3
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1, 1, 2, 1, 1, 5, 8, 5, 1, 1, 9, 29, 42, 29, 9, 1, 1, 14, 75, 196, 268, 196, 75, 14, 1, 1, 20, 160, 660, 1519, 2000, 1519, 660, 160, 20, 1, 1, 27, 301, 1800, 6299, 13293, 17038, 13293, 6299, 1800, 301, 27, 1, 1, 35, 518, 4235, 21000, 65485, 129681, 162890, 129681, 65485, 21000, 4235, 518, 35, 1, 1, 44, 834, 8932, 59633, 258720, 740046, 1395504, 1725372, 1395504, 740046, 258720, 59633, 8932, 834, 44, 1, 1, 54, 1275, 17316, 149787, 863982, 3386879, 9054684, 16420458, 20044728, 16420458, 9054684, 3386879, 863982, 149787, 17316, 1275, 54, 1
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n} T(n,k)*y^k satisfies
(1) A(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + (k+2)*y + y^2).
(2) A(x,y) = 1/(1 - x*y)^((1+y)^2/y).
(3) x = Sum_{n>=1} (x/A(x))^n/n! * Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2) ).
Row sums are (n+3)!/3! for row n >= 0.
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EXAMPLE
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E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n} T(n,k)*y^k starts
A(x,y) = 1 + (y^2 + 2*y + 1)*x + (y^4 + 5*y^3 + 8*y^2 + 5*y + 1)*x^2/2! + (y^6 + 9*y^5 + 29*y^4 + 42*y^3 + 29*y^2 + 9*y + 1)*x^3/3! + (y^8 + 14*y^7 + 75*y^6 + 196*y^5 + 268*y^4 + 196*y^3 + 75*y^2 + 14*y + 1)*x^4/4! + (y^10 + 20*y^9 + 160*y^8 + 660*y^7 + 1519*y^6 + 2000*y^5 + 1519*y^4 + 660*y^3 + 160*y^2 + 20*y + 1)*x^5/5! + (y^12 + 27*y^11 + 301*y^10 + 1800*y^9 + 6299*y^8 + 13293*y^7 + 17038*y^6 + 13293*y^5 + 6299*y^4 + 1800*y^3 + 301*y^2 + 27*y + 1)*x^6/6! + (y^14 + 35*y^13 + 518*y^12 + 4235*y^11 + 21000*y^10 + 65485*y^9 + 129681*y^8 + 162890*y^7 + 129681*y^6 + 65485*y^5 + 21000*y^4 + 4235*y^3 + 518*y^2 + 35*y + 1)*x^7/7! + (y^16 + 44*y^15 + 834*y^14 + 8932*y^13 + 59633*y^12 + 258720*y^11 + 740046*y^10 + 1395504*y^9 + 1725372*y^8 + 1395504*y^7 + 740046*y^6 + 258720*y^5 + 59633*y^4 + 8932*y^3 + 834*y^2 + 44*y + 1)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y) begins:
1;
1, 2, 1;
1, 5, 8, 5, 1;
1, 9, 29, 42, 29, 9, 1;
1, 14, 75, 196, 268, 196, 75, 14, 1;
1, 20, 160, 660, 1519, 2000, 1519, 660, 160, 20, 1;
1, 27, 301, 1800, 6299, 13293, 17038, 13293, 6299, 1800, 301, 27, 1;
1, 35, 518, 4235, 21000, 65485, 129681, 162890, 129681, 65485, 21000, 4235, 518, 35, 1;
1, 44, 834, 8932, 59633, 258720, 740046, 1395504, 1725372, 1395504, 740046, 258720, 59633, 8932, 834, 44, 1;
1, 54, 1275, 17316, 149787, 863982, 3386879, 9054684, 16420458, 20044728, 16420458, 9054684, 3386879, 863982, 149787, 17316, 1275, 54, 1; ...
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PROG
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(PARI) {T(n, k) = polcoeff( prod(m=0, n-1, 1 + (m+2)*y + y^2 +x*O(x^k)), k, y)}
for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))
(PARI) {T(n, k) = n!*polcoeff(polcoeff( 1/(1 - x*y +x*O(x^n) )^((1+y)^2/y), n, x), k, y)}
for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))
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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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