A054563 -id:A054563 - cp's OEIS frontend

A373733 Triangular numbers with tetrahedral indices: a(n) = (1/72) * (n(n+1)(n+2)(n+3)(n^2+2)).

0, 1, 10, 55, 210, 630, 1596, 3570, 7260, 13695, 24310, 41041, 66430, 103740, 157080, 231540, 333336, 469965, 650370, 885115, 1186570, 1569106, 2049300, 2646150, 3381300, 4279275, 5367726, 6677685, 8243830, 10104760, 12303280

Offset: 0

J.S. Seneschal, Jun 17 2024

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000292, A000217, A054563.

PolygonalNumber[Binomial[Range[2, 50], 3]] (* Paolo Xausa, Jul 22 2024 *)

a(n)=n*(n+1)*(n+2)*(n+3)*(n^2+2)/72 \\ Charles R Greathouse IV, Jul 10 2024

from math import comb
def A373733(n): return comb(comb(n+2,3)+1,2) # Chai Wah Wu, Jul 10 2024

a(n) = A000217(A000292(n)).

a(n) = A000292(n) + A054563(n).

A054644 Number of labeled pure 2-complexes on n nodes with 3 2-simplexes.

Original entry on oeis.org

4, 120, 1140, 6545, 27720, 95284, 280840, 735130, 1750540, 3858140, 7971964, 15596035, 29112720, 52174360, 90223760, 151173044, 246274580, 391222160, 607525380, 924205205, 1379864024, 2025189100, 2925954200, 4166590350

Offset: 4

Vladeta Jovovic, Apr 15 2000

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A054563.

[n*(n-1)*(n-2)*(n-3)*(n^2+2)*(n^3-3*n^2+2*n-12)/1296: n in [4..30]]; // Vincenzo Librandi, Apr 30 2012

Table[Binomial[Binomial[n,3],3],{n,4,60}] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{4,120,1140,6545,27720,95284,280840,735130,1750540,3858140},30] (* Vincenzo Librandi, Apr 30 2012 *)

[(binomial(binomial(n,3),3)) for n in range(4, 28)] # Zerinvary Lajos, Nov 30 2009

a(n) = binomial(binomial(n, 3), 3) = 4*binomial(n, 4) + 100*binomial(n, 5) + 480*binomial(n, 6) + 945*binomial(n, 7) + 840*binomial(n, 8) + 280*binomial(n, 9) = n*(n-1)*(n-2)*(n-3)*(n^2+2)*(n^3 - 3*n^2 + 2*n - 12)/1296.

G.f.: x^4*(4 + 80*x + 120*x^2 + 65*x^3 + 10*x^4 + x^5)/(1-x)^10. - Colin Barker, Jan 19 2012

More terms from James Sellers, Apr 16 2000

A093567 Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).

Original entry on oeis.org

0, 1, 14, 75, 265, 735, 1736, 3654, 7050, 12705, 21670, 35321, 55419, 84175, 124320, 179180, 252756, 349809, 475950, 637735, 842765, 1099791, 1418824, 1811250, 2289950, 2869425, 3565926, 4397589, 5384575, 6549215, 7916160, 9512536

Offset: 2

Robert G. Wilson v and Santi Spadaro, Mar 31 2004

nonn

All terms are positive: A093566 >= A054563 ==> C( C(n,2), 3) >= C( C(n,3), 2) ==> n^2*(n^4 + 3n^3 -35n^2 + 69n -38)/144 >= 0 ==> (n - 2)(n - 1)(n^2 + 6n - 19) ==> 0 which it is for all n >= 2.

Solomon W. Golomb, Iterated binomial coefficients, Amer. Math. Monthly, 87 (1980), 719-727.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A054563, A093566.

A093567:=n->binomial(binomial(n, 2), 3) - binomial(binomial(n, 3), 2); seq(A093567(n), n=2..30); # Wesley Ivan Hurt, Feb 02 2014

Table[ Binomial[ Binomial[n, 2], 3] - Binomial[ Binomial[n, 3], 2], {n, 2, 34}]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,14,75,265,735,1736},40] (* Harvey P. Dale, Jun 12 2016 *)

a(n) = binomial(binomial(n,2), 3) - binomial(binomial(n,3), 2); \\ Michel Marcus, Oct 01 2017

a(n) = A093566(n) - A054563(n).

G.f.: x^3*(-1-7*x+2*x^2+x^3)/(x-1)^7. - R. J. Mathar, Dec 08 2010

cp's OEIS Frontend

A373733 Triangular numbers with tetrahedral indices: a(n) = (1/72) * (n(n+1)(n+2)(n+3)(n^2+2)).

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A054644 Number of labeled pure 2-complexes on n nodes with 3 2-simplexes.

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A093567 Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).

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cp's OEIS Frontend

A373733 Triangular numbers with tetrahedral indices: a(n) = (1/72) * (n*(n+1)*(n+2)*(n+3)*(n^2+2)).

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Author

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Mathematica

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Python

Formula

A054644 Number of labeled pure 2-complexes on n nodes with 3 2-simplexes.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A093567 Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A373733 Triangular numbers with tetrahedral indices: a(n) = (1/72) * (n(n+1)(n+2)(n+3)(n^2+2)).