A245334 -id:A245334 - cp's OEIS frontend

A002690 a(n) = (n+1) * (2*n)! / n!.

1, 4, 36, 480, 8400, 181440, 4656960, 138378240, 4670265600, 176432256000, 7374868300800, 337903056691200, 16838835658444800, 906706535454720000, 52459449551308800000, 3245491278907637760000, 213796737998040637440000, 14940619102451310428160000, 1103945744792235714969600000

Offset: 0

N. J. A. Sloane

Coefficients of orthogonal polynomials.
E.g.f. for series with alternating signs: x/(1+4*x)^(1/2).
Central terms of triangle A245334. - Reinhard Zumkeller, Aug 30 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177.
H. E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. 9 (1955), 164-177. [Annotated scanned copy]

a(n) = (n+1) * A001813(n) = 2^n * A001193(n+1).
Cf. A002691, A000407.
Cf. A245334.

a002690 n = a245334 (2 * n) n  -- Reinhard Zumkeller, Aug 30 2014

[(n+1) * Factorial(2*n) /Factorial(n): n in [0..20]]; // Vincenzo Librandi, Sep 05 2011

with(combstruct):bin := {B=Union(Z,Prod(B,B))}:
seq (count([B,bin,labeled],size=n+1)*(n+1), n=0..17); # Zerinvary Lajos, Dec 05 2007
A002690 := n -> 2^n*n!*JacobiP(n, -1/2, -n+1, 3):
seq(simplify(A002690(n)), n = 0..18);  # Peter Luschny, Jan 22 2025

Table[((n+1)(2n)!)/n!,{n,0,20}] (* Harvey P. Dale, Sep 04 2011 *)

PARI
```
a(n)=(n+1)*(2*n)!/n!
```

E.g.f.: (1-2*x)/(1-4*x)^(3/2).

a(n) = 2^n*n!*JacobiP(n, -1/2, -n+1, 3). - Peter Luschny, Jan 22 2025

Edited by Ralf Stephan, Mar 21 2004

A056001 a(n) = (n+1)*binomial(n+7, 7).

Original entry on oeis.org

1, 16, 108, 480, 1650, 4752, 12012, 27456, 57915, 114400, 213928, 381888, 655044, 1085280, 1744200, 2728704, 4167669, 6229872, 9133300, 13156000, 18648630, 26048880, 35897940, 48859200, 65739375, 87512256, 115345296, 150629248, 195011080, 250430400, 319159632

Offset: 0

Barry E. Williams, Jun 18 2000

Original name: A second-order recursive sequence.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Partial sums of A052226.
Cf. A093565 ((8, 1) Pascal, column m=8).
Cf. A007318, A000142, A245334.

List([0..30], n-> (n+1)*Binomial(n+7,7)); # G. C. Greubel, Aug 29 2019

a056001 n = (n + 1) * a007318' (n + 7) 7
-- Reinhard Zumkeller, Aug 31 2014

[(n+1)*Binomial(n+7,7): n in [0..30]]; // G. C. Greubel, Aug 29 2019

seq((n+1)*binomial(n+7,7), n=0..30); # G. C. Greubel, Aug 29 2019

Table[(n+1)Binomial[n+7, 7], {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011; corrected by Bruno Berselli, Jan 23 2015 *)

vector(30, n, n*binomial(n+6,7)) \\ G. C. Greubel, Aug 29 2019

[(n+1)*binomial(n+7,7) for n in (0..30)] # G. C. Greubel, Aug 29 2019

G.f.: (1+7*x)/(1-x)^9.
a(n) = A245334(n+7,7)/A000142(7). - Reinhard Zumkeller, Aug 31 2014
a(n) = A000581(n+8)+7*A000581(n+7). - R. J. Mathar, Oct 24 2014
E.g.f.: (5040 +75600*x +194040*x^2 +170520*x^3 +66150*x^4 +12642*x^5 + 1225*x^6 +57*x^7 +x^8)*exp(x)/5040. - G. C. Greubel, Aug 29 2019
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 7*Pi^2/6 - 37583/3600.
Sum_{n>=0} (-1)^n/a(n) = 7*Pi^2/12 - 2912*log(2)/15 + 155701/1200. (End)

A056114 Expansion of (1+9*x)/(1-x)^11.

Original entry on oeis.org

1, 20, 165, 880, 3575, 12012, 35035, 91520, 218790, 486200, 1016158, 2015520, 3821090, 6963880, 12257850, 20920064, 34730575, 56241900, 89049675, 138138000, 210315105, 314757300, 463681725, 673171200, 964177500, 1363732656, 1906401420, 2636011840, 3607704980

Offset: 0

Barry E. Williams, Jun 12 2000

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000142, A007318, A056003, A245334.

List([0..40], n-> (n+1)*Binomial(n+9, 9)); # G. C. Greubel, Jan 18 2020

a056114 n = (n + 1) * a007318' (n + 9) 9
-- Reinhard Zumkeller, Aug 31 2014

[(n+1)*Binomial(n+9, 9): n in [0..40]]; // G. C. Greubel, Jan 18 2020

a:=n->(sum((numbcomp(n,10)), j=10..n)):seq(a(n), n=10..34); # Zerinvary Lajos, Aug 26 2008

CoefficientList[Series[(1+9x)/(1-x)^11,{x,0,40}],x] (* or *) LinearRecurrence[ {11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,20,165,880,3575,12012,35035, 91520,218790,486200,1016158},40] (* Harvey P. Dale, Jun 05 2018 *)

vector(41, n, n*binomial(n+8, 9) ) \\ G. C. Greubel, Jan 18 2020

[(n+1)*binomial(n+9, 9) for n in (0..40)] # G. C. Greubel, Jan 18 2020

a(n) = (n+1)*binomial(n+9, 9).
G.f.: (1+9*x)/(1-x)^11.
a(n) = A245334(n+9,9)/A000142(9). - Reinhard Zumkeller, Aug 31 2014
From G. C. Greubel, Jan 18 2020: (Start)
a(n) = 10*binomial(n+10,10) - 9*binomial(n+9,9).
E.g.f.: (9! +6894720*x +22861440*x^2 +26853120*x^3 +14605920*x^4 + 4191264*x^5 +677376*x^6 +63072*x^7 +3321*x^8 +91*x^9 +x^10)*exp(x)/9!. (End)
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 3*Pi^2/2 - 1077749/78400.
Sum_{n>=0} (-1)^n/a(n) = 3*Pi^2/4 - 24576*log(2)/35 + 37652469/78400. (End)

A192849 Molecular topological indices of the triangular graphs.

Original entry on oeis.org

0, 0, 24, 240, 1080, 3360, 8400, 18144, 35280, 63360, 106920, 171600, 264264, 393120, 567840, 799680, 1101600, 1488384, 1976760, 2585520, 3335640, 4250400, 5355504, 6679200, 8252400, 10108800, 12285000, 14820624, 17758440, 21144480

Offset: 1

Eric W. Weisstein, Jul 11 2011

Triangular graphs are defined for n>=2; extended to n=1 using closed form.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. D. Birkhoff, A determinant formula for the number of ways of coloring a map, Ann. Math., 14:42-4. See 2nd polynomial p. 5.
Eric Weisstein's World of Mathematics, Molecular Topological Index.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A027800, A245334.

List([1..40], n -> n*(n^2 -1)*(n-2)^2); # G. C. Greubel, Jan 05 2019

a192849 n = if n < 3 then 0 else a245334 (n + 1) 4
-- Reinhard Zumkeller, Aug 31 2014

[n*(n^2 -1)*(n-2)^2: n in [1..40]]; // G. C. Greubel, Jan 05 2019

[n*(n^2-1)*(n-2)^2$n=1..40]; # Muniru A Asiru, Jan 05 2019

Table[n*(n^2-1)*(n-2)^2, {n,1,40}] (* G. C. Greubel, Jan 05 2019 *)

vector(40, n, n*(n^2 -1)*(n-2)^2) \\ G. C. Greubel, Jan 05 2019

[n*(n^2 -1)*(n-2)^2 for n in (1..40)] # G. C. Greubel, Jan 05 2019

a(n) = n*(n^2 - 1)*(n-2)^2.
a(n) = 24*A027800(n-3).
G.f.: 24*x^3*(4*x+1)/(x-1)^6. - Colin Barker, Aug 07 2012
a(n) = A245334(n+1,4), n > 2. - Reinhard Zumkeller, Aug 31 2014
E.g.f.: x^3*(4 + 6*x + x^2)*exp(x). - G. C. Greubel, Jan 05 2019
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=3} 1/a(n) = Pi^2/36 - 49/216.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/72 - 10*log(2)/9 + 145/216. (End)

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A002690 a(n) = (n+1) * (2*n)! / n!.

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A056001 a(n) = (n+1)*binomial(n+7, 7).

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A056114 Expansion of (1+9*x)/(1-x)^11.

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A192849 Molecular topological indices of the triangular graphs.

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GAP

Haskell

Magma

Maple

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PARI

Sage

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