A151989 -id:A151989 - cp's OEIS frontend

A234043 a(n) = binomial(5*(n+1),4)/5, with n >= 0.

1, 42, 273, 969, 2530, 5481, 10472, 18278, 29799, 46060, 68211, 97527, 135408, 183379, 243090, 316316, 404957, 511038, 636709, 784245, 956046, 1154637, 1382668, 1642914, 1938275, 2271776, 2646567, 3065923, 3533244, 4052055, 4626006, 5258872, 5954553, 6717074

Offset: 0

Wolfdieter Lang, Feb 24 2014

Used as one of the 5-section parts of A234042.

The Fuss-Catalan numbers are Cat(d,k) = (1/(k*(d-1)+1))*binomial(k*d,k) and enumerate the (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link). a(n) = Cat(n,5) (Offset=1), so enumerates the (n+1)-gon partitions of a (5*(n-1)+2)-gon. Analogous series are A000326 (k=3) and A100157 (k=4). - Tom Copeland, Oct 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.

Cf. A100157, A000326, A001622, A151989, A234042.

[Binomial(5*(n+1),4)/5: n in [0..40]]; // Vincenzo Librandi, Feb 26 2014

CoefficientList[Series[(1 + 37 x + 73 x^2 + 14 x^3)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)

G.f: (1 + 37*x + 73*x^2 + 14*x^3)/(1-x)^5.
a(n) = A234042(5*n+1) for n >= 0.
a(n) = (n+1)*(5*n+2)*(5*n+3)*(5*n+4)/24.
From Amiram Eldar, Sep 20 2022: (Start)
Sum_{n>=0} 1/a(n) = 10*sqrt(5)*log(phi) + 5*log(5) - 2*sqrt(25-38/sqrt(5))*Pi, where phi is the golden ratio (A001622).
Sum_{n>=0} (-1)^n/a(n) = 4*sqrt(5)*log(phi) + 2*sqrt(26-38/sqrt(5))*Pi - 32*log(2). (End)

A362863 Centered hecatonicosachoral numbers.

Original entry on oeis.org

1, 1441, 11521, 44641, 122401, 273601, 534241, 947521, 1563841, 2440801, 3643201, 5243041, 7319521, 9959041, 13255201, 17308801, 22227841, 28127521, 35130241, 43365601, 52970401, 64088641, 76871521, 91477441, 108072001, 126828001, 147925441, 171551521, 197900641

Offset: 1

Léo Cymrot Cymbalista, May 06 2023

A hecatonicosachoral number is a centered figurate number that represents a hecatonicosachoron, which is a four-dimensional regular polytope composed of 120 cells.

One of the 6 centered regular polichoral (centered pentachoral, centered hexadecachoral, centered octachoral, centered icositetrachoral, centered hexacosichoral and centered hecatonicosachoral) numbers.

Eric Weisstein's World of Mathematics, Figurate Number.
Wikipedia, 120-cell.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005891 (2D), A005904 (3D), A006322, A151989.

Table[300*n^4 - 600*n^3 + 420*n^2 - 120*n + 1, {n, 1, 100}]

a(n) = 300*n^4 - 600*n^3 + 420*n^2 - 120*n + 1.
a(n) = 1440*A006322(n-1) + 1 for n > 1.
a(n) = 288*(A151989(n-1)-1)/25 + 1.
G.f.: x*(1 + 1436*x + 4326*x^2 + 1436*x^3 + x^4)/(1 - x)^5. - Stefano Spezia, May 12 2023

A001512 a(n) = (5n+1)(5n+2)(5n+3)(5*n+4).

Original entry on oeis.org

24, 3024, 24024, 93024, 255024, 570024, 1113024, 1974024, 3258024, 5085024, 7590024, 10923024, 15249024, 20748024, 27615024, 36060024, 46308024, 58599024, 73188024, 90345024, 110355024, 133518024, 160149024, 190578024, 225150024, 264225024, 308178024

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001622, A151989.

[(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4): n in[0..50]] // Vincenzo Librandi, Aug 02 2010

Table[Times@@(5n+{1,2,3,4}),{n,0,30}] (* Harvey P. Dale, Jul 15 2019 *)

a(n)=(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4) \\ Charles R Greathouse IV, Oct 21 2022

G.f.: 24*( x^4 + 121*x^3 + 381*x^2 + 121*x + 1 )/( 1-x )^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 [corrected by Jaume Oliver Lafont, Sep 19 2009]
From Amiram Eldar, Sep 20 2022: (Start)
Sum_{n>=0} 1/a(n) = sqrt(10-22/sqrt(5))*Pi/30.
Sum_{n>=0} (-1)^n/a(n) = 4*log(2)/15 - 2*log(phi)/(3*sqrt(5)), where phi is the golden ratio (A001622). (End)

A238471 a(n) = binomial(5n+6, 4)/5 for n >= 0.

Original entry on oeis.org

3, 66, 364, 1197, 2990, 6293, 11781, 20254, 32637, 49980, 73458, 104371, 144144, 194327, 256595, 332748, 424711, 534534, 664392, 816585, 993538, 1197801, 1432049, 1699082, 2001825, 2343328, 2726766, 3155439, 3632772, 4162315, 4747743, 5392856, 6101579, 6877962

Offset: 0

Wolfdieter Lang, Feb 28 2014

This sequence appears in the 5-section of A234042.

Cf. A000332, A001622, A234042, A151989, A234043, A238472, A238473.

a[n_] := Binomial[5*n + 6, 4]/5; Array[a, 40, 0] (* Amiram Eldar, Sep 20 2022 *)

a(n) = binomial(5*n+6, 4)/5 = (5*n+6)*(5*n+3)*(5*n+4)*(n+1)/4! for n >= 0.
a(n) = A234042(5*n+2) for n >= 0.
a(n) = 3*b(n) + 51*b(n-1) + 64*b(n-2) + 7*b(n-3), with b(n) = binomial(n+4,4) = A000332(n) for n >= 0.
O.g.f.: (3 + 51*x + 64*x^2 + 7*x^3)/(1-x)^5.
Sum_{n>=0} 1/a(n) = 2*sqrt(5+2/sqrt(5))*Pi - 10*sqrt(5)*log(phi) - 15*log(5) + 20, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 20 2022

A238472 a(n) = binomial(5*n+7, 4)/5 for n >= 0.

Original entry on oeis.org

7, 99, 476, 1463, 3510, 7192, 13209, 22386, 35673, 54145, 79002, 111569, 153296, 205758, 270655, 349812, 445179, 558831, 692968, 849915, 1032122, 1242164, 1482741, 1756678, 2066925, 2416557, 2808774, 3246901, 3734388, 4274810, 4871867, 5529384, 6251311, 7041723

Offset: 0

Wolfdieter Lang, Feb 28 2014

This sequence appears in the 5-section of A234042.

Cf. A000332, A001622, A234042, A151989, A234043, A238471, A238473.

a[n_] := Binomial[5*n + 7, 4]/5; Array[a, 40, 0] (* Amiram Eldar, Sep 20 2022 *)

a(n) = binomial(5*n+7, 4)/5 for n >= 0.
a(n) = A234042(5*n+3) for n >= 0.
a(n) = 7*b(n) + 64*b(n-1) + 51*b(n-2) + 3*b(n-3), with b(n) = binomial(n+4,4) = A000332(n) for n >= 0.
O.g.f.: (7 + 64*x + 51*x^2 + 3*x^3)/(1-x)^5.
Sum_{n>=0} 1/a(n) = 2*sqrt(5+2/sqrt(5))*Pi + 10*sqrt(5)*log(phi) + 15*log(5) - 50, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 20 2022

A238473 a(n) = binomial(5*n+8, 4)/5 for n >= 0.

Original entry on oeis.org

14, 143, 612, 1771, 4095, 8184, 14763, 24682, 38916, 58565, 84854, 119133, 162877, 217686, 285285, 367524, 466378, 583947, 722456, 884255, 1071819, 1287748, 1534767, 1815726, 2133600, 2491489, 2892618, 3340337, 3838121, 4389570, 4998409, 5668488, 6403782, 7208391

Offset: 0

Wolfdieter Lang, Feb 28 2014

This sequence appears in the 5-section of A234042.

Cf. A000332, A001622, A234042, A151989, A234043, A238471, A238472.

a[n_] := Binomial[5*n + 8, 4]/5; Array[a, 40, 0] (* Amiram Eldar, Sep 20 2022 *)

a(n) = binomial(5*n+8, 4)/5 = (5*n+8)*(5*n+7)*(5*n+6)*(n+1)/4! for n >= 0.
a(n) = A234042(5*n+2) for n >= 0.
a(n) = 14*b(n) + 73*b(n-1) + 37*b(n-2) + b(n-3), with b(n) = binomial(n+4,4) = A000332(n) for n >= 0.
O.g.f.: (14 + 73*x + 37*x^2 + x^3)/(1-x)^5.
Sum_{n>=0} 1/a(n) = 110/3 - 2*sqrt(25 - 38/sqrt(5))*Pi - 10*sqrt(5)*log(phi) - 5*log(5), where phi is the golden ratio (A001622). - Amiram Eldar, Sep 20 2022

A205795 Sums of coefficients of polynomials from 5n-th moments of X ~ Hypergeometric(4m, 5m, m).

Original entry on oeis.org

24, 2880, 43545600, 5230697472000, 2432902008176640000, 3102242008666197196800000, 8841761993739701954543616000000, 49205466506600690141269768273920000000, 485663859076129603777149565235783270400000000, 7911522544013240381082219675638737768808448000000000

Offset: 1

John M. Campbell, Feb 09 2012

nonn

See Maple code below for formula for such polynomials.

			The evaluation of sum(binomial(n, k)*binomial(4*n, k)*k^5, k = 0 .. n) involves the polynomial  256*n^5-640*n^3+400*n^2+108*n-100, the sum of the coefficients of which is 24 = a(1).

Eric W. Weisstein, MathWorld: Binomial Sums
Wikipedia, Hypergeometric Distribution
Index to divisibility sequences

Cf. A204820, A203778, A015219, A202948, A202946.

with(PolynomialTools);polyn:=w->simplify(Pi^2*sum(binomial(n,k)*binomial(4*n,k)*k^w,k=0..n)*5^w/3125^n*csc((1/5)*Pi)*csc((2/5)*Pi)*GAMMA(4*n)/GAMMA(n-(floor((w+1)/5)*5-2)/5)/GAMMA(n-(floor(w/5)*5-1)/5)/GAMMA(n-(floor((w+2)/5)*5-3)/5)/GAMMA(n-(floor((w+3)/5)*5-4)/5));coefl:=d->CoefficientList(expand(polyn(d)),n);seq(sum(coefl(5*h)[m],m=1..nops(coefl(5*h))),h=1..5);seq(simplify(12*5^(5*n-5)*GAMMA(n-4/5)*GAMMA(n-3/5)*GAMMA(n-2/5)*GAMMA(n-1/5)*(cos((2/5)*Pi)+cos((1/5)*Pi))/Pi^2),n=1..5);

a(n) = 120*A151989(n-2)*a(n-1), with a(1)=24.

a(n) = 12*5^(5*n-5)*GAMMA(n-4/5)*GAMMA(n-3/5)*GAMMA(n-2/5)*GAMMA(n-1/5)*(cos((2/5)*Pi)+cos((1/5)*Pi))/Pi^2.

cp's OEIS Frontend

A234043 a(n) = binomial(5*(n+1),4)/5, with n >= 0.

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A362863 Centered hecatonicosachoral numbers.

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A001512 a(n) = (5*n+1)*(5*n+2)*(5*n+3)*(5*n+4).

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A238471 a(n) = binomial(5n+6, 4)/5 for n >= 0.

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A238472 a(n) = binomial(5*n+7, 4)/5 for n >= 0.

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A238473 a(n) = binomial(5*n+8, 4)/5 for n >= 0.

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A205795 Sums of coefficients of polynomials from 5n-th moments of X ~ Hypergeometric(4m, 5m, m).

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A001512 a(n) = (5n+1)(5n+2)(5n+3)(5*n+4).