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A102036
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Triangle, read by rows, where the terms are generated by the rule: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-3,k-1), with T(0,0)=1.
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2
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1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 9, 15, 7, 1, 1, 12, 33, 28, 9, 1, 1, 15, 60, 81, 45, 11, 1, 1, 18, 96, 189, 161, 66, 13, 1, 1, 21, 141, 378, 459, 281, 91, 15, 1, 1, 24, 195, 675, 1107, 946, 449, 120, 17, 1, 1, 27, 258, 1107, 2349, 2673, 1742, 673, 153, 19, 1
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OFFSET
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0,5
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COMMENTS
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Row sums form A077939. This sequence was inspired by Luke Hanna.
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LINKS
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FORMULA
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T(n,k) = Sum_{m=0..(n-k)} Sum_{j=0..k}(C(j,m-j)*C(k,j))*C(n-m,k)). - Vladimir Kruchinin, Apr 21 2015
G.f. of column k: Sum_{n>=0} T(n+k,k) * x^n = (1+x+x^2)^k / (1-x)^(k+1) = (1-x^3)^k / (1-x)^(2*k+1).
Let k >= 0 be some fixed integer and a_k(n) be multiplicative with a_k(p^e) = T(e+k,k) for prime p and e >= 0. Then we have the Dirichlet g.f.: Sum{n>0} a_k(n) / n^s = (zeta(s))^(2*k+1) / (zeta(3*s))^k. (End)
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EXAMPLE
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Generated by adding preceding terms in the triangle at positions that form the letter 'L':
T(n,k) =
T(n-3,k-1) +
T(n-2,k-1) +
T(n-1,k-1) + T(n-1,k).
Rows begin:
[1],
[1, 1],
[1, 3, 1],
[1, 6, 5, 1],
[1, 9, 15, 7, 1],
[1, 12, 33, 28, 9, 1],
[1, 15, 60, 81, 45, 11, 1],
[1, 18, 96, 189, 161, 66, 13, 1],
[1, 21, 141, 378, 459, 281, 91, 15, 1], ...
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MAPLE
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T:=(n, k)->add(add((binomial(j, m-j)*binomial(k, j))*binomial(n-m, k), j=0..k), m=0..n-k): seq(seq(T(n, k), k=0..n), n=0..10); # Muniru A Asiru, Dec 11 2018
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MATHEMATICA
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T[n_, k_] := If[n < k || k < 0, 0, If[n == 0, 1, T[n - 1, k] + T[n - 1, k - 1] + T[n - 2, k - 1] + T[n - 3, k - 1]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 07 2018 *)
Table[Sum[Binomial[n-m, k]*Sum[Binomial[j, m-j]*Binomial[k, j], {j, 0, k}], {m, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 11 2018 *)
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PROG
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(PARI) {T(n, k)=if(n<k||k<0, 0, if(n==0, 1, T(n-1, k)+T(n-1, k-1)+T(n-2, k-1)+T(n-3, k-1)))}
(Maxima) T(n, k):=sum((sum(binomial(j, m-j)*binomial(k, j), j, 0, k))*binomial(n-m, k), m, 0, n-k); /* Vladimir Kruchinin, Apr 21 2015 */
(Magma) [[(&+[Binomial(n-m, k)*(&+[Binomial(j, m-j)*Binomial(k, j):j in [0..k]]): m in [0..n-k]]): k in [0..n]]: n in [0..15]]; // G. C. Greubel, Dec 11 2018
(Sage) [[sum(binomial(n-m, k)*sum(binomial(j, m-j)*binomial(k, j) for j in (0..k)) for m in (0..n-k)) for k in (0..n)] for n in range(15)] # G. C. Greubel, Dec 11 2018
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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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