cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A276482 a(n) = 5^n*Gamma(n+1/5)*Gamma(n+1)/Gamma(1/5).

Original entry on oeis.org

1, 1, 12, 396, 25344, 2661120, 415134720, 90084234240, 25944259461120, 9573431741153280, 4403778600930508800, 2470519795122015436800, 1660189302321994373529600, 1316530116741341538208972800, 1216473827868999581305090867200, 1295544626680484554089921773568000
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 05 2016

Keywords

Comments

12-gonal (or dodecagonal) factorial numbers, also polygorial(n, 12).
More generally, the m-gonal factorial numbers (or polygorial(n, m)) is 2^(-n)*(m - 2)^n*Gamma(n+2/(m-2))*Gamma(n+1)/Gamma(2/(m-2)), m>2.

Links

Crossrefs

Cf. A000142, A008548, A051624.
Cf. similar sequences of m-gonal factorial numbers (or polygorial(n, m)): A006472 (m=3), A001044 (m=4), A084939 (m=5), A000680 (m=6), A084940 (m=7), A084941 (m=8), A084942 (m=9), A084943 (m=10), A084944 (m=11).

Programs

  • Maple
    seq(mul(k*(5*k-4),k=1..n), n=0..20); # Robert Israel, Sep 18 2016
  • Mathematica
    FullSimplify[Table[5^n Gamma[n + 1/5] (Gamma[n + 1]/Gamma[1/5]), {n, 0, 15}]]
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2),n]]; Array[polygorial[12, #] &, 16, 0] (* Robert G. Wilson v, Dec 13 2016 *)
  • PARI
    a(n) = prod(k=1, n, k*(5*k - 4)); \\ Michel Marcus, Sep 06 2016

Formula

a(n) = Product_{k=1..n} k*(5*k - 4), a(0)=1.
a(n) = Product_{k=1..n} A051624(k), a(0)=1.
a(n) = A000142(n)*A008548(n).
a(n) ~ 2*Pi*5^n*n^(2*n+1/5)/(Gamma(1/5)*exp(2*n)).
Sum_{n>=0} 1/a(n) = BesselI(-4/5,2/sqrt(5))*Gamma(1/5)/5^(2/5) = Hypergeometric0F1(1/5, 1/5) = 2.085898421130914...

A279662 a(n) = (2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2)/Gamma(3/4).

Original entry on oeis.org

1, 1, 7, 154, 7700, 731500, 117771500, 29678418000, 11040371496000, 5796195035400000, 4144279450311000000, 3920488359994206000000, 4790836775912919732000000, 7411424492337286825404000000, 14266992147749277138902700000000, 33670101468688294047810372000000000
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 16 2016

Keywords

Comments

Hexagonal pyramidal factorial numbers.
More generally, the m-gonal pyramidal factorial numbers is 6^(-n)*(m-2)^n*Gamma(n+1)*Gamma(n+2)*Gamma(n+3/(m-2))/Gamma(3/(m-2)), m>2.

Links

Crossrefs

Cf. A002412.
Cf. A000680 (hexagonal factorial numbers).
Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279663 (heptagonal pyramidal factorial numbers).

Programs

  • Magma
    [Round((2/3)^n*Gamma(n+3/4)*Gamma(n+1)*Gamma(n+2) / Gamma(3/4)): n in [0..20]]; // Vincenzo Librandi, Dec 17 2016
  • Mathematica
    FullSimplify[Table[(2/3)^n Gamma[n + 3/4] Gamma[n + 1] Gamma[n + 2]/Gamma[3/4], {n, 0, 15}]]

Formula

a(n) = Product_{k=1..n} k*(k + 1)*(4*k - 1)/6, a(0)=1.
a(n) = Product_{k=1..n} A002412(k), a(0)=1.
a(n) ~ (2*Pi)^(3/2)*(2/3)^n*n^(3*n+9/4)/(Gamma(3/4)*exp(3*n)).

A318259 Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, 5, -11, 6, -61, 211, -240, 90, 1385, -6551, 11466, -8820, 2520, -50521, 303271, -719580, 844830, -491400, 113400, 2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400, -199360981, 1704396331, -6187282920, 12372329970, -14727913200, 10443232800, -4086482400, 681080400
Offset: 0

Views

Author

Peter Luschny, Sep 06 2018

Keywords

Comments

The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See the cross-references for some other members.
The unsigned numbers have row sums A210657 which points to an interpretation of the unsigned numbers as a refinement of marked Schröder paths (see Josuat-Vergès and Kim).

Examples

			[0] [      1]
[1] [     -1,         1]
[2] [      5,       -11,        6]
[3] [    -61,       211,     -240,        90]
[4] [   1385,     -6551,    11466,     -8820,     2520]
[5] [ -50521,    303271,  -719580,    844830,  -491400,    113400]
[6] [2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400]

Links

Crossrefs

Row sums are A000007, alternating row sums are A210657.
Cf. T(n,n) = A000680, T(n, 0) = A028296(n) (Gudermannian), A000364 (Euler secant), A241171 (Joffe's differences), A028246 (Worpitzky).
Cf. A167374 (m=0), A028246 & A163626 (m=1), this seq (m=2), A318260 (m=3).

Programs

  • Maple
    Joffe := proc(n, k) option remember; if k > n then 0 elif k = 0 then k^n else
k*(2*k-1)*Joffe(n-1, k-1)+k^2*Joffe(n-1, k) fi end:
T := (n, k) -> add((-1)^(k-j)*binomial(n-j, n-k)*add((-1)^i*Joffe(n,i)*
binomial(n-i, j), i=0..n), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..6);
  • Mathematica
    Joffe[0, 0] = 1; Joffe[n_, k_] := Joffe[n, k] = If[k>n, 0, If[k == 0,k^n, k*(2*k-1)*Joffe[n-1, k-1] + k^2*Joffe[n-1, k]]];
T[n_, k_] := Sum[(-1)^(k-j)*Binomial[n-j, n-k]*Sum[(-1)^i*Joffe[n, i]* Binomial[n-i, j], {i, 0, n}], {j, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2019, from Maple *)
  • Sage
    def EW(m, n):
    @cached_function
    def S(m, n):
        R. = ZZ[]
        if n == 0: return R(1)
        return R(sum(binomial(m*n, m*k)*S(m, n-k)*x for k in (1..n)))
    s = S(m, n).list()
    c = lambda k: sum((-1)^(k-j)*binomial(n-j,n-k)*
        sum((-1)^i*s[i]*binomial(n-i,j) for i in (0..n)) for j in (0..k))
    return [c(k) for k in (0..n)]
def A318259row(n): return EW(2, n)
flatten([A318259row(n) for n in (0..6)])

Formula

Let S(n, k) denote Joffe's central differences of zero (A241171) extended to the case n = 0 and k = 0 by prepending a column 1, 0, 0, 0,... to the triangle, then:
T(n,k) = Sum_{j=0..k}((-1)^(k-j)*C(n-j,n-k)*Sum_{i=0..n}((-1)^i*S(n,i)*C(n-i,j))).

A362582 Triangular array read by rows. T(n,k) is the number of alternating permutations of [2n+1] having exactly 2k elements to the left of 1, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 5, 6, 5, 61, 75, 75, 61, 1385, 1708, 1750, 1708, 1385, 50521, 62325, 64050, 64050, 62325, 50521, 2702765, 3334386, 3427875, 3438204, 3427875, 3334386, 2702765, 199360981, 245951615, 252857605, 253708455, 253708455, 252857605, 245951615, 199360981
Offset: 0

Views

Author

Geoffrey Critzer, Apr 25 2023

Keywords

Comments

Here, w = w_1,w_2,...,w_(2n+1) is an alternating permutation if w_1 < w_2 > w_3 < ... < w_(2n) > w_(2n+1).

Examples

			T(2,1) = 6 because we have: {2, 3, 1, 5, 4}, {2, 4, 1, 5, 3}, {2, 5, 1, 4, 3}, {3, 4, 1, 5, 2}, {3, 5, 1, 4, 2}, {4, 5, 1, 3, 2}.
Triangle begins
     1;
     1,     1;
     5,     6,     5;
    61,    75,    75,    61;
  1385,  1708,  1750,  1708,  1385;
 50521, 62325, 64050, 64050, 62325, 50521;
 ...

Links

Crossrefs

Cf. A000182 (row sums), A000364 (column k=0), A000680.
Cf. A000111, A104345.

Programs

  • Maple
    b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
T:= (n, k)-> binomial(2*n, 2*k)*b(2*k, 0)*b(2*(n-k), 0):
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Apr 25 2023
  • Mathematica
    nn = 6; B[n_] := (2 n)!/2^n; e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[1/e[-u z]*1/e[-z], {z, 0, nn}], {z, u}]] // Grid

Formula

Sum_{n>=0} Sum_{k=0..n} T(n,k)*u^k*z^n/A000680(n) = 1/(E(-u*z)*E(-z)) where E(z) = Sum_{n>=0} z^n/A000680(n).
T(n,k) = binomial(2*n,2*k)*A000111(2*k)*A000111(2*(n-k)). - Alois P. Heinz, Apr 25 2023

A370086 Expansion of e.g.f. exp( Sum_{k>=1} (2*k)!/k! * (x/2)^k/k ).

Original entry on oeis.org

1, 1, 4, 40, 796, 27196, 1437136, 108931264, 11207616400, 1502077491856, 254091983968576, 52922687300496256, 13303823750214614464, 3970706309867394765760, 1387875547214097148600576, 561507863501525383223535616, 260328000140228961840632033536
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2024

Keywords

Crossrefs

Cf. A000680, A370084.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, (2*k)!/k!*(x/2)^k/k))))

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} (2*k)!/2^k * binomial(n,k) * a(n-k).

A154284 A triangle sequence of polynomial coefficients: p(x,n)=(x - 1)^(3*n + 1)*Sum[(k*(k + 1)*(2*k + 1)/6)^n*x^k, {k, 0, Infinity}]/x.

Original entry on oeis.org

1, 1, -1, -18, -42, -18, -1, 1, 115, 1539, 5065, 5065, 1539, 115, 1, -1, -612, -30369, -359056, -1439038, -2255448, -1439038, -359056, -30369, -612, -1, 1, 3109, 487944, 16069256, 177275075, 808273143, 1688579472, 1688579472, 808273143
Offset: 1

Views

Author

Roger L. Bagula, Jan 06 2009

Keywords

Comments

Row sums are:
{2, -80, 13440, -5913600, 5381376000, -8782405632000, 23361198981120000,
-94566133475573760000, 553211880832106496000000, -4492080472356704747520000000,...}

Examples

			{1, 1},
{-1, -18, -42, -18, -1},
{1,115, 1539, 5065, 5065, 1539, 115, 1},
{-1, -612, -30369, -359056, -1439038, -2255448, -1439038, -359056, -30369, -612, -1},
{1, 3109,487944, 16069256, 177275075, 808273143, 1688579472, 1688579472, 808273143, 177275075, 16069256,487944, 3109, 1},
{-1, -15606, -7232832, -588609722, -15102054532, -159360510654, -796011579264, -2034786608786, -2770692409206, -2034786608786, -796011579264, -159360510654, -15102054532, -588609722, -7232832, -15606, -1},
{1, 78103, 103694985, 19568948247, 1065525448614, 23072731441362, 236032579067166, 1262043871882890, 3749020958436984, 6409344151561648, 6409344151561648, 3749020958436984, 1262043871882890, 236032579067166, 23072731441362, 1065525448614, 19568948247, 103694985, 78103, 1},
{-1, -390600, -1466023731, -619322458800, -67773276182575, -2802617455410216, -53645573041228725, -536366226569480256, -3023314553367761850, -10090695137544912400, -20563892762682272046, -26024562946121517600, -20563892762682272046, -10090695137544912400, -3023314553367761850, -536366226569480256, -53645573041228725, -2802617455410216, -67773276182575, -619322458800, -1466023731, -390600, -1},
{1, 1953097, 20606359662, 19105228968022, 4062046061251702, 306352064179097622, 10355782284092172382, 180449348295691590742, 1772697944064120724647, 10422061778244020252047, 38179816523099760064252, 89512894147925375525772, 136527354458904110040052, 136527354458904110040052, 89512894147925375525772, 38179816523099760064252, 10422061778244020252047, 1772697944064120724647, 180449348295691590742, 10355782284092172382, 306352064179097622, 4062046061251702, 19105228968022, 20606359662, 1953097, 1},
{-1, -9765594, -288951921066, -581527646706874, -235124431637251555, -31362739743718489620, -1800493681167699729940, -52143903931162149522580, -843767280698373213383505, -8174015411024678270125030, -49739731928304174504355510, -196644863528988544778420550, -517044213844787972256968395, -918147140681329965091033240, -1110785055014320687664653080, -918147140681329965091033240, -517044213844787972256968395, -196644863528988544778420550, -49739731928304174504355510, -8174015411024678270125030, -843767280698373213383505, -52143903931162149522580, -1800493681167699729940, -31362739743718489620, -235124431637251555, -581527646706874, -288951921066, -9765594, -1}

Crossrefs

Cf. A000680.

Programs

  • Mathematica
    Clear[p, x, n]; p[x_, n_] = (x - 1)^(3*n + 1)*Sum[(k*(k + 1)*(2*k + 1)/6)^n*x^k, {k, 0, Infinity}]/x;
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]

Formula

p(x,n)=(x - 1)^(3*n + 1)*Sum[(k*(k + 1)*(2*k + 1)/6)^n*x^k, {k, 0, Infinity}]/x;
t(n,m)=Coefficients(p(x,n)).

A211309 a(n) = number |fdw(P,(n))| of entangled P-words with s=2.

Original entry on oeis.org

1, 4, 60, 1776, 84720, 5876640, 556466400, 68882446080
Offset: 1

Views

Author

N. J. A. Sloane, Apr 08 2012

Keywords

Comments

See Jenca and Sarkoci for the precise definition.

Links

Crossrefs

Cf. A000680, A000698, A001147.

Formula

From Peter Bala, Sep 05 2012: (Start)
Conjectural e.g.f.: 2 - 1/A(x), where A(x) = sum {n = 0..inf} (2*n)!/2^n*x^n/n! is the e.g.f. for A000680 (also the o.g.f. for A001147).
If true, this gives a(n) = n!*A000698(n) and leads to the recurrence equation: a(n) = (2*n)!/2^n - sum {k = 1..n-1} (2*k)!/2^k*binomial(n,k)*a(n-k) with a(1) = 1.
(End)

A256880 n*n!/round(n/2).

Original entry on oeis.org

1, 4, 9, 48, 200, 1440, 8820, 80640, 653184, 7257600, 73180800, 958003200, 11564467200, 174356582400, 2451889440000, 41845579776000, 671854030848000, 12804747411456000, 231125690776780800, 4865804016353280000
Offset: 1

Views

Author

M. F. Hasler, Apr 21 2015

Keywords

Crossrefs

Cf. A000142, A052145.

Programs

Formula

a(2n-1) = A052145(n).
a(2n) = 4*A002674(n) = 2*A010050(n) = 2^(n+1)*A000680(n), n>=1.

A264153 a(n) = ((2*n)!)^2 / 2^n.

Original entry on oeis.org

1, 2, 144, 64800, 101606400, 411505920000, 3585039575040000, 59375425441812480000, 1710012252724199424000000, 80059353648041568632832000000, 5780285333388601255290470400000000, 616883611349898303167109582028800000000, 93983451956379706284115479041251737600000000
Offset: 0

Views

Author

Peter Luschny, Nov 06 2015

Keywords

Crossrefs

Cf. A000079, A000680, A134372, A264152.

Programs

  • Maple
    a := n -> (2*n)!^2/2^n; seq(a(n), n=0..10);
  • Sage
    a = lambda n: factorial(2*n)^2 >> n
[a(n) for n in range(11)]

Formula

a(n) = A134372(n)/A000079(n).
a(n)*A264152(n) = A134372(n)*A006882(2*n-1)/A006882(n).
a(n)/A264152(n) is an integer: 1, 1, 24, 1620,....

A269943 Triangle read by rows, T(n,k) = ((-1)^k*(2*n)!/4^k)*P[n,k](1/((2*n-1)*(2*n))) where P is the inverse P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 6, 0, 16, 60, 90, 0, 288, 1176, 2520, 2520, 0, 9216, 39360, 98280, 151200, 113400, 0, 460800, 2023296, 5504400, 10311840, 12474000, 7484400, 0, 33177600, 148442112, 426666240, 896575680, 1362160800, 1362160800, 681080400
Offset: 0

Views

Author

Peter Luschny, Mar 27 2016

Keywords

Comments

The P-transform is defined in the link. Compare also the Sage implementation below.

Examples

			Triangle starts:
[1]
[0, 1]
[0, 2, 6]
[0, 16, 60, 90]
[0, 288, 1176, 2520, 2520]
[0, 9216, 39360, 98280, 151200, 113400]
[0, 460800, 2023296, 5504400, 10311840, 12474000, 7484400]

Links

Crossrefs

Cf. A000680, A055546.

Programs

  • Sage
    # uses[PtransMatrix from A269941]
eul = lambda n: 1/((2*n-1)*(2*n))
norm = lambda n,k: (-1)^k*factorial(2*n)/4^k
PtransMatrix(7, eul, norm, inverse=True)

Formula

T(n,1) = 2^(n-1)*(n-1)!^2 (cf. A055546) for n>=1.
T(n,n) = (2*n)!/2^n = A000680(n).
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