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.

Showing 1-7 of 7 results.

A259462 From higher-order arithmetic progressions.

Original entry on oeis.org

1, 30, 1200, 70000, 5880000, 691488000, 110638080000, 23471078400000, 6454546560000000, 2256222608640000000, 985518035453952000000, 529939925428193280000000, 346227417946419609600000000, 271655358696421539840000000000, 253338025938605687439360000000000, 278215820085776765945905152000000000, 356811789260008702325623357440000000000
Offset: 0

Views

Author

N. J. A. Sloane, Jun 30 2015

Keywords

Comments

"3 over n!" in Dienger's article is A087047. A_1 is A000217. - Georg Fischer, Dec 16 2024

Links

Crossrefs

Cf. A087047, A259463, A259464.

Programs

  • Maple
    rXI := proc(n, a, d)
        n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
end proc:
A259462 := proc(n)
        mul(rXI(i, a, d), i=1..n+1) ;
        coeftayl(%, d=0, 1) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259462(n), n=1..25) ; # R. J. Mathar, Jul 15 2015
  • Mathematica
    rXI[n_, a_, d_] := n(n+1)(n+2)/6*a + (n+2)(n+1)n(n-1)/24*d;
A259462[n_] :=
   Product[rXI[i, a, d], {i, 1, n + 2}] //
   SeriesCoefficient[#, {d, 0, 1}] & //
   SeriesCoefficient[#, {a, 0, n + 1}] & ;
Table[A259462[n], {n, 0, 14}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

Formula

D-finite with recurrence: -6*n*a(n) +(n+4)*(n+3)*(n+2)^2*a(n-1)=0. - R. J. Mathar, Jul 15 2015
a(n) = 2^(-n-3)*3^(-n-2)*(n+2)!*(n+3)!*(n+4)!/4*(n+2)*(n+1)/2. - Georg Fischer, Dec 16 2024

A259463 From higher-order arithmetic progressions.

Original entry on oeis.org

5, 550, 61250, 8330000, 1440600000, 318084480000, 88994505600000, 31196975040000000, 13537335651840000000, 7186069008518400000000, 4614893517270516480000000, 3548831033950800998400000000, 3237226357799023349760000000000, 3472842105575052965314560000000000
Offset: 0

Views

Author

N. J. A. Sloane, Jun 30 2015

Keywords

Comments

"3 over n!" on page 15 in the Dienger article is A087047; A_2 is A000914. - Georg Fischer, Dec 16 2024

Links

Crossrefs

Cf. A000914, A087047, A259462, A259464.

Programs

  • Maple
    rXI := proc(n, a, d)
        n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
end proc:
A259463 := proc(n)
        mul(rXI(i, a, d), i=1..n+2) ;
        coeftayl(%, d=0, 2) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259463(n), n=1..25) ; # R. J. Mathar, Jul 15 2015
  • Mathematica
    rXI[n_, a_, d_] := n(n+1)(n+2)/6 a + (n+2)(n+1)n(n-1)/24 d;
A259463[n_] := Product[rXI[i, a, d], {i, 1, n+3}]//
   SeriesCoefficient[#, {d, 0, 2}]&//
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259463[n], {n, 0, 13}] (* Jean-François Alcover, May 02 2023, after R. J. Mathar *)

Formula

D-finite with recurrence: -6*n*(3*n+5)*a(n) +(n+5)*(n+4)*(3*n+8)*(n+3)^2*a(n-1)=0. - R. J. Mathar, Jul 15 2015
a(n) = 2^(-n-4)*3^(-n-3)*(n+3)!*(n+4)!*(n+5)!*(n+2)*(n+1)*(n+3)*(3*n+8)/384. - Georg Fischer, Dec 16 2024

Extensions

Corrected by Jean-François Alcover, May 02 2023

A259464 From higher-order arithmetic progressions.

Original entry on oeis.org

75, 21875, 5512500, 1512630000, 484041600000, 184834742400000, 84715923600000000, 46534591303200000000, 30489464221856640000000, 23681690417572387200000000, 21660852835272876825600000000, 23175597788788462617600000000000, 28817200450516396946227200000000000
Offset: 0

Views

Author

N. J. A. Sloane, Jun 30 2015

Keywords

Comments

"3 over n!" on page 15 in the Dienger article is A087047; A_3 is A001303. - Georg Fischer, Dec 16 2024

Links

Crossrefs

Cf. A001303, A087047, A259462, A259463.

Programs

  • Maple
    rXI := proc(n, a, d)
        n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
end proc:
A259464 := proc(n)
        mul(rXI(i, a, d), i=1..n+3) ;
        coeftayl(%, d=0, 3) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259464(n), n=1..25) ; # R. J. Mathar, Jul 15 2015
  • Mathematica
    rXI[n_, a_, d_] := (n(n+1)(n+2)/6)*a+((n+2)(n+1)n(n-1)/24)*d;
A259464[n_] :=
   Product[rXI[i, a, d], {i, 1, n+4}]//
   SeriesCoefficient[#, {d, 0, 3}]&//
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259464[n], {n, 0, 12}] (* Jean-François Alcover, Apr 26 2023, after R. J. Mathar *)

Formula

D-finite with recurrence: -6*n*(n+2)*a(n) +(n+6)*(n+5)*(n+4)^3*a(n-1)=0. - R. J. Mathar, Jul 15 2015
a(n) = 2^(-n-5)*3^(-n-4)*(n+4)!*(n+5)!*(n+6)!*(n+4)^2*(n+3)^2*(n+2)*(n+1)/3072. - Georg Fischer, Dec 16 2024

A140729 Diagonal A(n,n) of array A(k,n) = Product of first n of k-gonal pyramidal numbers.

Original entry on oeis.org

40, 2100, 324000, 117771500, 86640153600, 115851776040000, 260111401804800000, 922852527136155000000, 4931966428685936640000000, 38193820496218904209973280000, 415101787718859995456102400000000
Offset: 3

Views

Author

Jonathan Vos Post, May 25 2008

Keywords

Comments

The array A(k,n) = Product of first n k-gonal pyramidal numbers begins:
===================================================================
..|n=1|n=2|..n=3|...n=4..|......n=5....|......n=6......|......n=7......|.......n=8.........|
k=3|.1.|.4.|..40.|....800.|.......28000.|.......1568000.|.....131712000.|.......15805440000.|A087047
k=4|.1.|.5.|..70.|...2100.|......115500.|......10510500.|....1471470000.|......300179880000.|
k=5|.1.|.6.|.108.|...4320.|......324000.|......40824000.|....8001504000.|.....2304433152000.|
k=6|.1.|.7.|.154.|...7700.|......731500.|.....117771500.|...29678418000.|....11040371496000.|
k=7|.1.|.8.|.208.|..12480.|.....1435200.|.....281299200.|...86640153600.|....39507910041600.|
k=8|.1.|.9.|.270.|.718900.|.....2551500.|.....589396500.|..214540326000.|...115851776040000.|
===================================================================

Examples

			a(3) = product of the first 3 triangular pyramidal (tetrahedral) numbers (A000292) = A087047(3) = 1 * 4 * 10 = 40.
a(4) = product of the first 4 square pyramidal numbers (A000330) = 1 * 5 * 14 * 30 = 2100.
a(5) = product of the first 5 pentagonal pyramidal numbers (A002411) = 1 * 6 * 18 * 40 * 75 = 324000.
a(6) = product of the first 6 hexagonal pyramidal numbers (A002412) = 1 * 7 * 22 * 50 * 95 * 161 = 117771500.
a(7) = product of the first 7 heptagonal pyramidal numbers (A002413) = 1 * 8 * 26 * 60 * 115 * 196 * 308 = 86640153600.
a(8) = product of the first 8 octagonal pyramidal numbers (A002414) = 1 * 9 * 30 * 70 * 135 * 231 * 364 * 540 = 115851776040000.

Links

Crossrefs

Cf. A000292, A000330, A002411, A002412, A002413, A002414, A140729.

Programs

  • Maple
    A130729 := proc(n) n!*(n+1)!*(n-2)^n*pochhammer(1+(5-n)/(n-2),n)/6^n ; end: seq(A130729(n),n=3..30) ; # R. J. Mathar, May 31 2008

Formula

A(k,n) = PRODUCT[j=1..n] (1/6)*j*(j+1)*[(k-2)*j+(5-k)].
a(n) ~ Pi^(3/2) * n^(4*n + 1/2) / (2^(n - 3/2) * 3^(n-1) * exp(3*n+2)) * (1 + (3*log(n) + 3*gamma + 5/4)/n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 29 2023

Extensions

More terms from R. J. Mathar, May 31 2008

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)).

A279663 a(n) = (5/6)^n*Gamma(n+3/5)*Gamma(n+1)*Gamma(n+2)/Gamma(3/5).

Original entry on oeis.org

1, 1, 8, 208, 12480, 1435200, 281299200, 86640153600, 39507910041600, 25482601976832000, 22424689739612160000, 26147188236387778560000, 39429959860472770068480000, 75350653293363463600865280000, 179334554838205043370059366400000, 523656900127558726640573349888000000
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 16 2016

Keywords

Comments

Heptagonal pyramidal factorial numbers.

Links

Crossrefs

Cf. A002413.
Cf. A084940 (heptagonal factorial numbers).
Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279662 (hexagonal pyramidal factorial numbers).

Programs

  • Magma
    [Round((5/6)^n*Gamma(n+3/5)*Gamma(n+1)*Gamma(n+2)/Gamma(3/5)): n in [0..20]]; // Vincenzo Librandi Dec 17 2016
  • Mathematica
    FullSimplify[Table[(5/6)^n Gamma[n + 3/5] Gamma[n + 1] Gamma[n + 2]/Gamma[3/5], {n, 0, 15}]]

Formula

a(n) = Product_{k=1..n} k*(k + 1)*(5*k - 2)/6, a(0)=1.
a(n) = Product_{k=1..n} A002413(k), a(0)=1.
a(n) ~ (2*Pi)^(3/2)*(5/6)^n*n^(3*n+21/10)/(Gamma(3/5)*exp(3*n)).

A381056 Product of row n of A329708.

Original entry on oeis.org

1, 16, 4320, 7680000, 56672000000, 1315328716800000, 79725223359774720000, 11041460968683995136000000, 3159164253667495772160000000000, 1725992749819407775039488000000000000, 1690274868390850110509130354524160000000000, 2816890048270042497343000411961733572198400000000
Offset: 0

Views

Author

Darío Clavijo, Feb 12 2025

Keywords

Examples

			Row n=2 of A329708 is {1, 4, 10, 12, 9} and the product of those is a(2) = 4320.

Crossrefs

Cf. A000290, A000292, A000537, A005408, A007531, A087047, A329708.

Programs

  • PARI
    a(n) = vecprod(Vec(sum(k=0, n, (k+1)*x^k)^2)); \\ Michel Marcus, Feb 13 2025
  • Python
    from sympy import prod
def a(n):
    p = ((n+1)*(n+2)*(n+3)) // 6
    p *= prod(((k*(k+1)*(k+2))*((n+k+1)*(n+k+2)*(n+k+3)-2*k*(k+1)*(3*n+k+5)))//36  for k in range(1,n+1))
    return p
print([a(n)  for n in range(0, 14)])

Formula

a(n) = Product_{k=0..2*n} A329708(n,k).
a(n) = Product_{k=0..2*n} (Sum_{i=max(0,k-n)..min(k,n)} (i+1)*(k-i+1)).
a(n) == 0 (mod (n+1)^3).
a(n) = (n+1)*(n+2)*(n+3)*(1/6)*(Product_{k=1..n} k*(k+1)*(k+2)*((n+k+1)*(n+k+2)*(n+k+3)-2*k*(k+1)*(3*n+k+5))/36).
a(n) = binomial(n+3,3)*(Product_{k=1..n} binomial(k+2,3)*(binomial(k+n+3,3)-(k*(k+1)*(3*n+k+5))/3)).
a(n) = (Product_{k=0..n-1} Sum_{i=0..k} (i+1)*(k-i+1)) * (Product_{k=n..2*n} Sum_{i=k-n..n} (i+1)*(k-i+1)).
a(n) = (Product_{k=0..n-1} binomial(k+3,3))*(Product{k=n..2*n} binomial(k+3,3)-(2*n+4)*binomial(k+2-n)+(2*n+2)*binomial(k+1-n)).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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