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A138177
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Triangle T(n,k) read by rows: number of k X k symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n, n>=1, 1<=k<=n.
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6
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1, 1, 2, 1, 4, 4, 1, 7, 15, 10, 1, 10, 36, 52, 26, 1, 14, 74, 176, 190, 76, 1, 18, 132, 460, 810, 696, 232, 1, 23, 222, 1060, 2705, 3756, 2674, 764, 1, 28, 347, 2180, 7565, 15106, 17262, 10480, 2620, 1, 34, 525, 4204, 19013, 51162, 83440, 80816, 42732, 9496, 1, 40
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OFFSET
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1,3
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COMMENTS
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T(n,k) is also the number of semistandard Young tableaux of size n whose entries span the interval 1..k. See also Gus Wiseman's comment in A138178. The T(4,2) = 7 semi-standard Young tableaux of size 4 spanning the interval 1..2 are:
11 122 112 111 1222 1122 1112
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LINKS
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FORMULA
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EXAMPLE
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Triangle T(n,k) begins:
1;
1, 2;
1, 4, 4;
1, 7, 15, 10;
1, 10, 36, 52, 26;
1, 14, 74, 176, 190, 76;
1, 18, 132, 460, 810, 696, 232;
1, 23, 222, 1060, 2705, 3756, 2674, 764;
...
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MAPLE
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gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
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MATHEMATICA
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gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); A[n_, k_] := Coefficient[ Series [gf[k], {x, 0, n+1}], x, n]; T[n_, k_] := Sum[(-1)^j*Binomial[k, j]*A[n, k-j], {j, 0, k}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)
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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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