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 12 results. Next

A367543 a(n) = Product_{i=1..n, j=1..n} (i^2 - i*j + j^2).

Original entry on oeis.org

1, 36, 777924, 51190934086656, 32435802373365731229926400, 483207398728525904876601066508152707481600, 350969035472356907726779584093506665415605824531908346799718400
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 22 2023

Keywords

Crossrefs

Cf. A203312, A324403, A367542.
Cf. A324426, A079478.

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 A367543(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 # Chai Wah Wu, Nov 22 2023

Formula

a(n) = A324426(n) / A079478(n).
a(n) ~ 3^(1/6) * Gamma(1/3)^2 * n^(2*n^2 - 1/3) / (2^(5/3) * Pi^(5/3) * exp(3*n^2 - (n^2 + n + 1/6)*Pi/sqrt(3))).

A203012 Vandermonde sequence using x^2 + xy + y^2 applied to (1,2,...,n).

Original entry on oeis.org

1, 7, 1729, 37616124, 135933424914924, 132432199651531695045312, 51437933151214684812682944045953088, 11056394929890243558409721156996503083526683082752, 1743892714865607005898689849291524734866677095031979100765833773056
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Examples

			a(1)=1
a(2)=1^2+1*2+2^2=7
a(3)=(1^2+1*2+2^2)(1^3+1*3+3^2)(2^2+2*3+3^2)=1729.

Crossrefs

Cf. A203312, A203475, A203673, A367542.

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}]          (* A203012 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203158 *)

Formula

a(n) ~ c * n^(n^2 - n - 5/6) * 3^(n*(3*n+1)/4) / exp(3*n^2/2 - n - n*(n+1)*Pi / (4*sqrt(3))), where c = sqrt(Gamma(1/3)) * 3^(5/24) * exp(Pi/(24*sqrt(3))) / (2^(7/6) * Pi^(7/6)) = 0.26001211479205772659823692637002123572622409280442625312217301129630097... - 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...

A368064 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4*i*j + j^2).

Original entry on oeis.org

1, 6, 24336, 870746557824, 1311726482483997806493696, 256433546267136937832915286844640487014400, 15678550451426175377500759401206644047210595564950427820202393600
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

Cf. A367543, A324403, A367542, A079478^2, A368067, A368065, A368066.

Programs

  • Mathematica
    Table[Product[i^2 + 4*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) ~ 2^((3+sqrt(3))*n*(n+1) + (sqrt(3)-1)/6) * 3^(3*n*(n+1) + 13/24) * n^(2*n^2 - 7/6) / (Gamma(1/3)^(1/2) * Gamma(1/4)^(1/3) * Pi^(7/12) * (1 + sqrt(3))^((6*n*(n+1) + 1)/sqrt(3) - 1/2) * exp(3*n^2)).
Showing 1-10 of 12 results. Next

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

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