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A341867
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Square array read by downward antidiagonals: T(m,n) = Sum_{i=0..m, j=0..n} binomial(m,i)*binomial(n,j)*binomial(i+j,i).
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2
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1, 2, 2, 4, 5, 4, 8, 12, 12, 8, 16, 28, 33, 28, 16, 32, 64, 86, 86, 64, 32, 64, 144, 216, 245, 216, 144, 64, 128, 320, 528, 664, 664, 528, 320, 128, 256, 704, 1264, 1736, 1921, 1736, 1264, 704, 256, 512, 1536, 2976, 4416, 5322, 5322, 4416, 2976, 1536, 512
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OFFSET
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0,2
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COMMENTS
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T(m,n) is the coefficient of x^m*y^n of 1/(1 - 2*x - 2*y + 3*x*y).
In general, define T_{s,t}(m,n) = Sum_{i=0..m, j=0..n} binomial(m,i)*binomial(n,j)*binomial(i+j,i)*s^i*t^j, then T_{s,t}(m,n) is the coefficient of x^m*y^n of 1/(1 - (1+s)*x - (1+t)*y + (1+s+t)*x*y).
T(m,n) is the coefficient of x^n of (2 - 3*x)^m/(1 - 2*x)^(m+1). In general, T_{s,t}(m,n) is the coefficient of x^n of ((1+t) - (1+s+t)*x)^m/(1 - (1+s)*x)^(m+1).
T(m,n) is odd if and only if m = n. Proof: T(m,n) == T_{-1,-1}(m,n) (mod 2). The RHS is the coefficient of x^m*y^n of 1/(1 - x*y), which is 1 if m = n and 0 otherwise.
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LINKS
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FORMULA
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T(0,n) = Sum_{k=0..n} binomial(n,k) = 2^n;
T(1,n) = Sum_{k=0..n} binomial(n,k) * (k+2) = (n+4)*2^(n-1);
T(2,n) = Sum_{k=0..n} binomial(n,k) * (k^2+7*k+8)/2 = (n^2+15*n+32)*2^(n-3);
T(3,n) = Sum_{k=0..n} binomial(n,k) * (k^3+15*k^2+56*k+48)/6 = (n^3+33*n^2+254*n+384)*2^(n-4)/3.
E.g.f.: Sum_{m,n>=0} T(m,n)*x^m*y^n/(m!*n!) = exp(2*x+2*y) * BesselI(0,2*sqrt(x*y)). In general, Sum_{m,n>=0} T_{s,t}(m,n)*x^m*y^n/(m!*n!) = exp((1+s)*x+(1+t)*y) * BesselI(0,2*sqrt(s*t*x*y)). Note that BesselI(0,2*sqrt(x)) = Sum_{k>=0} x^k/(k!)^2.
E.g.f. for m-th row: Sum_{n>=0} T(m,n)*x^n/n! = exp(2*x) * Sum_{k=0..m} (binomial(m,k)*2^(m-k)/k!) * x^k. In general, Sum_{n>=0} T_{s,t}(m,n)*x^n/n! = exp((1+s)*x) * Sum_{k=0..m} (binomial(m,k)*(1+t)^(m-k)/k!) * (s*t*x)^k.
Define P_n(x) = exp(-x) * d^n/dx^n (x^n*exp(x)), then Sum_{n>=0} T_{s,t}(m,n)*x^n/n! = exp((1+s)*x) * ((1+t)^m/m!) * P_m(s*t*x/(1+t)) if t != -1 and Sum_{n>=0} T_{s,t}(m,n)*x^n/n! = exp((1+s)*x) * (s*t*x)^m/m! if t = -1.
T(m, n) = Sum_{j=0..n} binomial(n, j)*hypergeom([j + 1, -m], [1], -1). - Peter Luschny, Nov 08 2021
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EXAMPLE
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Rows 0-7:
1, 2, 4, 8, 16, 32, 64, 128, ...
2, 5, 12, 28, 64, 144, 320, 704, ...
4, 12, 33, 86, 216, 528, 1264, 2976, ...
8, 28, 86, 245, 664, 1736, 4416, 10992, ...
16, 64, 216, 664, 1921, 5322, 14268, 37272, ...
32, 144, 528, 1736, 5322, 15525, 43620, 118980, ...
64, 320, 1264, 4416, 14268, 43620, 127905, 362910, ...
128, 704, 2976, 10992, 37272, 118980, 362910, 1067925, ...
...
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MATHEMATICA
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T[m_, n_] := Sum[Binomial[m, i] * Binomial[n, j] * Binomial[i + j, i], {i, 0, m}, {j, 0, n} ]; Table[T[m, n - m], {n, 0, 9}, {m, 0, n}] // Flatten (* Amiram Eldar, Nov 08 2021 *)
T[m_, n_] := Sum[Binomial[n, j] Hypergeometric2F1[j + 1, -m, 1, -1], {j, 0, n}];
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PROG
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(PARI) T(m, n) = sum(i=0, m, sum(j=0, n, binomial(m, i)*binomial(n, j)*binomial(i+j, i)))
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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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