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A243660
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Triangle read by rows: the x = 1+q Narayana triangle at m=2.
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4
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1, 3, 2, 12, 16, 5, 55, 110, 70, 14, 273, 728, 702, 288, 42, 1428, 4760, 6160, 3850, 1155, 132, 7752, 31008, 50388, 42432, 19448, 4576, 429, 43263, 201894, 395010, 418950, 259350, 93366, 18018, 1430, 246675, 1315600, 3010700, 3853696, 3010700, 1466080, 433160, 70720, 4862
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OFFSET
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1,2
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COMMENTS
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See Novelli-Thibon (2014) for precise definition.
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)*...*(x+2n+1) / (n! * (2n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 09 2022
The Maple code T(n,k) := binomial(3*n+1-k,n-k)*binomial(2*n,k-1)/n: with(sumtools): sumrecursion( (-1)^(k+1)*T(n,k)*binomial(x+3*n-k+1, 3*n-k+1), k, s(n) ); returns the recurrence 2*(2*n+1)*n^2*s(n) = (x+n)*(x+2*n)*(x+2*n+1)*s(n-1). The above observation follows from this. - Peter Bala, Oct 30 2022
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LINKS
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FORMULA
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T(n,k) = binomial(3*n+1-k,n-k) * binomial(2*n,k-1) / n.
More generally: T(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(m*n,k-1) / n, where m = 2.
Sum_{k=1..n} (-1)^k * T(n,k) = -1. (End)
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EXAMPLE
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Triangle begins:
1;
3, 2;
12, 16, 5;
55, 110, 70, 14;
273, 728, 702, 288, 42;
1428, 4760, 6160, 3850, 1155, 132;
...
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MATHEMATICA
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polrecip[P_, x_] := P /. x -> 1/x // Together // Numerator;
P[n_, m_] := Sum[Binomial[m n + 1, k] Binomial[(m+1) n - k, n - k] (1-x)^k x^(n-k), {k, 0, n}]/(m n + 1);
T[m_] := Reap[For[i=1, i <= 20, i++, z = polrecip[P[i, m], x] /. x -> 1+q; Sow[CoefficientList[z, q]]]][[2, 1]];
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PROG
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(PARI)
N(n, m)=sum(k=0, n, binomial(m*n+1, k)*binomial((m+1)*n-k, n-k)*(1-x)^k*x^(n-k))/(m*n+1);
T(m)=for(i=1, 20, z=subst(polrecip(N(i, m)), x, 1+q); print(Vecrev(z)));
(PARI) T(n, k) = binomial(3*n+1-k, n-k) * binomial(2*n, k-1) / n; \\ Andrew Howroyd, Nov 23 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Corrected example and a(22)-a(43) from Lars Blomberg, Jul 12 2017
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STATUS
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approved
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