A383453 -id:A383453 - cp's OEIS frontend

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

Offset: 1

N. J. A. Sloane, Jun 13 2014

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

			Triangle begins:
     1;
     3,    2;
    12,   16,    5;
    55,  110,   70,   14;
   273,  728,  702,  288,   42;
  1428, 4760, 6160, 3850, 1155,  132;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 8.

Row sums give A034015(n-1).
The case m=1 is A126216 or A033282 (its mirror image).
The case m=3 is A243661.
The right diagonal is A000108.
The left column is A001764.
Same table as A383453, transposed. - Dan Eilers, May 06 2025

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]];
T[2] // Flatten (* Jean-François Alcover, Oct 08 2018, from 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)));
T(2)  /* Lars Blomberg, Jul 17 2017 */

T(n,k) = binomial(3*n+1-k,n-k) * binomial(2*n,k-1) / n; \\ Andrew Howroyd, Nov 23 2018

From Werner Schulte, Nov 23 2018: (Start)
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)

Corrected example and a(22)-a(43) from Lars Blomberg, Jul 12 2017

a(44)-a(45) from Werner Schulte, Nov 23 2018

A383440 a(n) = (5n + 3)!/((8n^2 + 10n + 3)(n!)^2(3n + 2)!).

Original entry on oeis.org

1, 16, 702, 42432, 3010700, 235282320, 19615029280, 1712821144320, 154870831986156, 14388837044278400, 1366276815032189060, 132069279628944665280, 12957831870375876372252, 1287484157116598357029120, 129316124278441748161584000, 13111175417326191857901849600

Offset: 0

Peter Luschny, May 03 2025

nonn

N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025), p. 17.

Cf. A383453, A383439.

a := n -> ((5*n + 3)!/((8*n^2 + 10*n + 3)*(n!)^2*(3*n + 2)!)):

Array[(5*# + 3)!/((8*#^2 + 10*# + 3)*(#!)^2*(3*# + 2)!) &, 16, 0] (* Michael De Vlieger, May 03 2025 *)

a(n) = A383453(2*n, n), conjectured by Wildberger-Rubine to be the main diagonal of the Geode Bi-Tri array G[m_2, m_3].

a(n) ~ 3^(-3*n-5/2)*5^(5*n+7/2)/(16*n^2*Pi). - Stefano Spezia, May 03 2025

A383456 a(n) = 2binomial(2n+4,n)*(n+3)/(n+2).

Original entry on oeis.org

3, 16, 70, 288, 1155, 4576, 18018, 70720, 277134, 1085280, 4249388, 16640960, 65189475, 255487680, 1001800650, 3930284160, 15427730010, 60591880800, 238098549780, 936100651200, 3682173186510, 14490838080576, 57053452670100, 224729967000448, 885568808835900, 3491063797713856, 13767643081368792, 54315062063687040, 214354554086311331

Offset: 0

N. J. A. Sloane, May 02 2025

Conjectured to be column 1 of the array in A383453.

Cf. A383453.

cp's OEIS Frontend

A383454 Row 1 of the array in A383453.

Views

Author

Keywords

Crossrefs

A383457 Column 2 of the array in A383453.

Views

Author

Keywords

Crossrefs

A383455 Row 2 of the array in A383453.

Views

Author

Keywords

Crossrefs

A243660 Triangle read by rows: the x = 1+q Narayana triangle at m=2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A383440 a(n) = (5*n + 3)!/((8*n^2 + 10*n + 3)*(n!)^2*(3*n + 2)!).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A383456 a(n) = 2*binomial(2*n+4,n)*(n+3)/(n+2).

Views

Author

Keywords

Comments

Crossrefs

A383440 a(n) = (5n + 3)!/((8n^2 + 10n + 3)(n!)^2(3n + 2)!).

A383456 a(n) = 2binomial(2n+4,n)*(n+3)/(n+2).