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A110582
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Triangle, read by rows, where the g.f. of diagonal n, D_n(x), is generated from the g.f. of row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2 for n>0, with D_0(x) = 1/(1-x)^2.
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1
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1, 1, 2, 1, 2, 3, 1, 4, 3, 4, 1, 4, 7, 4, 5, 1, 6, 10, 10, 5, 6, 1, 6, 14, 16, 13, 6, 7, 1, 8, 18, 26, 22, 16, 7, 8, 1, 8, 25, 34, 38, 28, 19, 8, 9, 1, 10, 29, 52, 55, 50, 34, 22, 9, 10, 1, 10, 37, 66, 84, 76, 62, 40, 25, 10, 11, 1, 12, 44, 90, 116, 122, 97, 74, 46, 28, 11, 12
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OFFSET
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0,3
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COMMENTS
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Related to planar partitions.
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..k} T(n-k-1, j)*(k-j+1) with T(n, n) = n+1.
G.f.: A(x, y) = Sum_{j=0..n} x^j/Product_{i=1..j+1} (1-y*x^i)^2.
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EXAMPLE
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Triangle begins:
1;
1,2;
1,2,3;
1,4,3,4;
1,4,7,4,5;
1,6,10,10,5,6;
1,6,14,16,13,6,7;
1,8,18,26,22,16,7,8;
1,8,25,34,38,28,19,8,9;
1,10,29,52,55,50,34,22,9,10; ...
{1,3,6,12,21,38,63,106,170,272,422,653,...},
(planar partitions with only one row and one column).
G.f. of diagonal n, D_n(x), is generated from g.f. of
row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2:
D_3(x) = 1 + 4*x + 10*x^2 + 16*x^3 + 22*x^4 + ...
= (1 + 2*x + 3*x^2)/(1-x)^2 = R_2(x)/(1-x)^2;
D_4(x) = 1 + 6*x + 14*x^2 + 26*x^3 + 38*x^4 + ...
= (1+ 4*x+ 3*x^2+ 4*x^3)/(1-x)^2 = R_3(x)/(1-x)^2.
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[n < k || k < 0, 0, If[k == 0, 1, If[k == n, n + 1, Sum[T[n - k - 1, j]*(k - j + 1), {j, 0, k}]]]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 01 2017 *)
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PROG
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(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, 1, if(k==n, n+1, sum(j=0, k, T(n-k-1, j)*(k-j+1))); ))
(PARI) T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); if(n<k || k<0, 0, polcoeff(polcoeff(sum(j=0, n, X^j/prod(i=1, j+1, 1-Y*X^i)^2), n, x), k, y))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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