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 1-10 of 13 results. Next

A324426 a(n) = Product_{i=1..n, j=1..n} (i^3 + j^3).

Original entry on oeis.org

1, 2, 2592, 134425267200, 3120795915109442519040000, 180825857777547616919759624941086965760000000, 99356698720512072045648926659510730227553351200000695922065408000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2019

Keywords

Crossrefs

Cf. A079478, A324403, A324437, A324438, A324439, A324440, A367834.
Cf. A272246, A307210, A367517, A367543.

Programs

  • Maple
    a:= n-> mul(mul(i^3+j^3, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Product[i^3+j^3, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
  • PARI
    a(n) = prod(i=1, n, prod(j=1, n, i^3+j^3)); \\ Michel Marcus, Feb 27 2019
  • Python
    from math import prod, factorial
def A324426(n): return prod(i**3+j**3 for i in range(1,n) for j in range(i+1,n+1))**2*factorial(n)**3<Chai Wah Wu, Nov 26 2023

Formula

a(n) ~ A * 2^(2*n*(n+1) + 1/4) * exp(Pi*(n*(n+1) + 1/6)/sqrt(3) - 9*n^2/2 - 1/12) * n^(3*n^2 - 3/4) / (3^(5/6) * Pi^(1/6) * Gamma(2/3)^2), where A is the Glaisher-Kinkelin constant A074962.
a(n) = A079478(n) * A367543(n). - Vaclav Kotesovec, Nov 22 2023
For n>0, a(n)/a(n-1) = A272246(n)^2 / (2*n^9). - Vaclav Kotesovec, Dec 02 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A367542 a(n) = Product_{i=1..n, j=1..n} (i^2 + i*j + j^2).

Original entry on oeis.org

3, 1764, 2905736652, 66016970246853190656, 64657853715047202043531429875379200, 6627957368676918780503749855130249245999452089509478400
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 22 2023

Keywords

Crossrefs

Cf. A203012, A324403, A367543.

Programs

  • Mathematica
    Table[Product[Product[(i^2 + i*j + j^2), {i, 1, n}], {j, 1, n}], {n, 1, 10}]
  • Python
    from math import prod, factorial
def A367542(n): return (prod(i*(i+j)+j**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2*3**n # Chai Wah Wu, Nov 22 2023

Formula

a(n) ~ c * 3^(3*n*(n+1)/2) * n^(2*n^2 - 2/3) / exp(3*n^2 - Pi*n*(n+1) / (2*sqrt(3))), where c = 3^(5/12) * exp(Pi/(12*sqrt(3))) * Gamma(1/3) / (2^(4/3) * Pi^(4/3)) = 0.42478290981890921418850643030484274341562970375995548434917...

A203312 Vandermonde sequence using x^2 - xy + y^2 applied to (1,2,...,n).

Original entry on oeis.org

1, 3, 147, 298116, 47460365316, 965460013501733568, 3717096745012192786213464768, 3763515081241454304168766426610670649344, 1329626784930718063722475681347135527472012731205697536
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Crossrefs

Cf. A203012, A203673, A367543.

Programs

  • Mathematica
    f[j_] := j; z = 12;
v[n_] := Product[Product[f[j]^2 - f[j] f[k] + f[k]^2,
{j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203312 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203513 *)
  • Python
    from operator import mul
from functools import reduce
def v(n): return 1 if n==1 else reduce(mul, [j**2 - j*k + k**2 for k in range(2, n + 1) for j in range(1, k)])
print([v(n) for n in range(1, 11)]) # Indranil Ghosh, Jul 26 2017

Formula

a(n) ~ c * n^(n^2 - n - 2/3) / exp(3*n^2/2 - n*(n+1)*Pi/(2*sqrt(3)) - n), where c = Gamma(1/3) * 3^(1/12) * exp(Pi/(12*sqrt(3))) / (2^(4/3) * Pi^(4/3)) = 0.2945280196744096322469352538791946777977998766871923997662057483092872... - Vaclav Kotesovec, Nov 22 2023

A367550 a(n) = Product_{i=1..n, j=1..n} (i^4 + i^2*j^2 + j^4).

Original entry on oeis.org

3, 63504, 2260442279270448, 3379470372507391964272022793486336, 2097229364987262298214192667129919538956418868293588090880000
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 22 2023

Keywords

Crossrefs

Cf. A203673, A324437, A367542, A367543, A367668.

Programs

  • Mathematica
    Table[Product[Product[i^4 + i^2*j^2 + j^4, {i, 1, n}], {j, 1, n}], {n, 1, 10}]
  • Python
    from math import prod, factorial
def A367550(n): return (prod((i2:=i**2)*(i2+(j2:=j**2))+j2**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n)**2)**2*3**n # Chai Wah Wu, Nov 22 2023

Formula

a(n) = A367542(n) * A367543(n).
a(n) ~ Gamma(1/3)^3 * 3^(3*n*(n+1)/2 + 7/12) * n^(4*n^2 - 1) / (8*Pi^3 * exp(6*n^2 - (6*n*(n+1) + 1)*Pi/(4*sqrt(3)))).

A368065 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*i*j + j^2).

Original entry on oeis.org

1, 7, 44100, 3210672937500, 12804360424787610000000000, 8591751256288909159255104643281750000000000, 2333034616280404811605303958158227652934766912996000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

Cf. A367543, A324403, A367542, A079478^2, A368067, A368064, A368066.
Cf. A368068, A368069.

Programs

  • Mathematica
    Table[Product[i^2 + 5*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) ~ c * 7^(7*n*(n+1)/2) * ((5-sqrt(21))/2)^(sqrt(21)*n*(n+1)/2) * n^(2*n^2 - 4/3) / exp(3*n^2), where c = A368069.

A367944 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*j^2).

Original entry on oeis.org

1, 6, 27216, 1344924798336, 3605580335899213007486976, 1648055031941075082958467426002632704000000, 312704667066499295437237787452750428210311485710262201221120000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2023

Keywords

Comments

In general, for d>0, Product_{i=1..n, j=1..n} (i^2 + d*j^2) ~ c(d) * n^(2*n^2 - 1/2) * (d+1)^(n*(n+1)) * d^(-n/2) * exp(n*(n+1)*(Pi*d/2 - (d-1)*arctan(sqrt(d))) / sqrt(d) - 3*n^2), where c(d) is a constant (dependent only on d).
c(1) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4), cf. A324403.

Crossrefs

Cf. A324403 (d=1), A367941 (d=2), A367942 (d=3), A367943 (d=4).
Cf. A367542, A367543, A367958.

Programs

  • Mathematica
    Table[Product[i^2+5*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

Formula

a(n) ~ c * n^(2*n^2 - 1/2) * 6^(n*(n+1)) * 5^(-n/2) * exp(n*(n+1)*(5*Pi/2 - 4*arctan(sqrt(5)))/sqrt(5) - 3*n^2), where c = 0.4431081869167792949266065295798218232844989957987096447783995373751372668...

A368066 a(n) = Product_{i=1..n, j=1..n} (i^2 + 6*i*j + j^2).

Original entry on oeis.org

1, 8, 73984, 10027173445632, 93867986947606492024406016, 185865459466664040069739311383413462872883200, 186896871826703385639703785281909582209471190408233074664996759142400
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Comments

In general, for d >= -1, Product_{i=1..n, j=1..n} (i^2 + d*i*j + j^2) ~ c(d) * (d+2)^((d+2)*n*(n+1)/2) * n^(2*n^2 - 1/2 - d/6) / ((d/2 + sqrt(d^2/4 - 1))^(sqrt(d^2 - 4)*n*(n+1)/2) * exp(3*n^2)), where c(d) is a constant (dependent only on d).
c(-1) = 3^(1/6) * exp(Pi/(6*sqrt(3))) * Gamma(1/3)^2 / (2*Pi)^(5/3).
c(0) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4).
c(1) = 3^(5/12) * exp(Pi/(12*sqrt(3))) * Gamma(1/3) / (2*Pi)^(4/3).
c(2) = A^2 / (2^(1/6) * exp(1/6) * Pi), where A = A074962.
c(3) = 2^((sqrt(5) - 9)/6) * sqrt(5) * (1 + sqrt(5))^(1/2 - sqrt(5)/6) / Pi.
c(4) = 2^((sqrt(3) - 1)/6) * 3^(13/24) * (1 + sqrt(3))^(1/2 - 1/sqrt(3)) / (Pi^(7/12) * Gamma(1/4)^(1/3) * Gamma(1/3)^(1/2)).
c(5) = A368069.
c(6) = 2^(25/8) * (1 + sqrt(2))^(3/4 - 2*sqrt(2)/3) / (Pi^(1/4) * Gamma(1/8) * Gamma(1/4)^(1/2)).
Special (non-integer) case: Product_{i=1..n, j=1..n} (i^2 + (d + 1/d)*i*j + j^2) ~ A^(2/d) * (Product_{j=1..d} Gamma(j/d)^(2*j/d)) * (d+1)^((d/2 + 1 + 1/(2*d))*2*n*(n+1) + (d+1)^2/(6*d) + 1/6) * n^(2*n^2 - d/6 - 1/2 - 1/(6*d)) / ((2*Pi)^((d+1)/2) * exp(3*n^2 + 1/(6*d)) * d^((d+1)*n*(n+1) - 1/(6*d))), where A = A074962 is the Glaisher-Kinkelin constant.

Crossrefs

Cf. A367543 (d=-1), A324403 (d=0), A367542 (d=1), A079478^2 (d=2), A368067 (d=3), A368064 (d=4), A368065 (d=5).
Cf. A368069, A368068.

Programs

  • Mathematica
    Table[Product[i^2 + 6*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) ~ 2^(12*n*(n+1) + 25/8) * n^(2*n^2 - 3/2) / (Pi^(1/4) * Gamma(1/4)^(1/2) * Gamma(1/8) * (1 + sqrt(2))^(2*sqrt(2)*(6*n*(n+1) + 1)/3 - 3/4) * exp(3*n^2)).

A367941 a(n) = Product_{i=1..n, j=1..n} (i^2 + 2*j^2).

Original entry on oeis.org

1, 3, 1944, 4102777008, 140890630179993255936, 247470977313135626800897828778803200, 54132901224855040835735917614114353691165557521593139200
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2023

Keywords

Crossrefs

Cf. A324403, A367942, A367943, A367944.
Cf. A367542, A367543.

Programs

  • Mathematica
    Table[Product[i^2+2*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

Formula

a(n) ~ c * n^(2*n^2 - 1/2) * 3^(n*(n+1)) * 2^(-n/2) * exp(n*(n+1)*(Pi - arctan(sqrt(2))) / sqrt(2) - 3*n^2) , where c = 0.4690673220228472212446336926899602910226601891141458824921925169726804439...

A367942 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3*j^2).

Original entry on oeis.org

1, 4, 5824, 45861064704, 9751658280030585225216, 176005320076923781520069562958715289600, 656508955366282248103393001602851493819854909361664242483200
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2023

Keywords

Crossrefs

Cf. A324403, A367941, A367943, A367944.
Cf. A367542, A367543.

Programs

  • Mathematica
    Table[Product[i^2+3*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

Formula

a(n) ~ c * n^(2*n^2 - 1/2) * 4^(n*(n+1)) * 3^(-n/2) * exp(5*Pi*n*(n+1)/(6*sqrt(3)) - 3*n^2), where c = 0.4612030005343304845802441101292774353695846313857765074861837886133930626...

A367943 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4*j^2).

Original entry on oeis.org

1, 5, 13600, 294372000000, 252880261890048000000000, 27099784799070466617992871936000000000000, 882065676199020188908312950703217787436793856000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 05 2023

Keywords

Crossrefs

Cf. A324403, A367941, A367942, A367944.
Cf. A367542, A367543.

Programs

  • Mathematica
    Table[Product[i^2+4*j^2, {i, 1, n}, {j, 1, n}], {n, 0, 8}]

Formula

a(n) ~ c * n^(2*n^2 - 1/2) * 5^(n*(n+1)) * 2^(-n) * exp(n*(n+1)*(2*Pi - 3*arctan(2))/2 - 3*n^2) , where c = 0.4523180383519335764034720087114905921141637339852374451758854101884791581...
Showing 1-10 of 13 results. Next

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