A010791 -id:A010791 - cp's OEIS frontend

A302913 Determinant of n X n matrix whose main diagonal consists of the first n 9-gonal numbers and all other elements are 1's.

1, 8, 184, 8280, 612720, 67399200, 10312077600, 2093351752800, 544271455728000, 176343951655872000, 69655860904069440000, 32947222207624845120000, 18384549991854663576960000, 11949957494705531325024000000, 8950518163534442962442976000000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

			The matrix begins:
1   1   1   1   1   1   1 ...
1   9   1   1   1   1   1 ...
1   1  24   1   1   1   1 ...
1   1   1  46   1   1   1 ...
1   1   1   1  75   1   1 ...
1   1   1   1   1 111   1 ...
1   1   1   1   1   1 154 ...

Cf. A001106 (nonagonal numbers).

Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), this sequence (k=9), A302914 (k=10).

d:=(i,j)->`if`(i<>j,1,i*(7*i-5)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);

nmax = 20; Table[Det[Table[If[i == j, i*(7*i-5)/2, 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
RecurrenceTable[{a[n+1] == a[n] * n*(7*n + 9)/2, a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[7^(n + 1) * Gamma[n] * Gamma[n + 9/7] / (9*Gamma[2/7]*2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(7*i-5)/2))); \\ Michel Marcus, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 7^(n+1) * Gamma(n) * Gamma(n + 9/7) / (9 * Gamma(2/7) * 2^n).
a(n) ~ Pi * 7^(n+1) * n^(2*n + 2/7) / (9 * Gamma(2/7) * 2^(n-1) * exp(2*n)).
a(n+1) = a(n) * n*(7*n + 9)/2.
(End)

A302914 Determinant of n X n matrix whose main diagonal consists of the first n 10-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 9, 234, 11934, 1002456, 125307000, 21803418000, 5036589558000, 1490830509168000, 550116457882992000, 247552406047346400000, 133430746859519709600000, 84861955002654535305600000, 62882708656967010661449600000, 53701833193049827104877958400000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

From Vaclav Kotesovec, Apr 16 2018: (Start)
In general, for k > 2, these determinants for k-gonal numbers satisfies:
a(n,k) = ((k-2)/2)^(n-1) * Gamma(n) * Gamma(n + k/(k-2)) / Gamma(1 + k/(k-2)).
a(n,k) ~ 4*Pi * (k/2 - 1)^n * n^(2*n + 2/(k-2)) / (k * Gamma(k/(k-2)) * exp(2*n)).
a(n+1,k) = a(n,k) * n*((k-2)*n + k)/2.
(End)

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1  10   1   1   1   1   1 ...
  1   1  27   1   1   1   1 ...
  1   1   1  52   1   1   1 ...
  1   1   1   1  85   1   1 ...
  1   1   1   1   1 126   1 ...
  1   1   1   1   1   1 175 ...

Cf. A001107.
Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), this sequence (k=10).
Cf. A007840 (permanent instead of determinant, for k=2).

d:=(i,j)->`if`(i<>j,1,i*(4*i-3)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);

nmax = 20; Table[Det[Table[If[i == j, i*(4*i-3), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
RecurrenceTable[{a[n+1] == a[n] * n*(4*n + 5), a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[4^(n+1) * Gamma[n] * Gamma[n + 5/4] / (5*Gamma[1/4])], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(4*i-3)))); \\ Michel Marcus, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 4^(n+1) * Gamma(n) * Gamma(n + 5/4) / (5*Gamma(1/4)).
a(n) ~ Pi * 2^(2*n + 3) * n^(2*n + 1/4) / (5 * Gamma(1/4) * exp(2*n)).
a(n+1) = a(n) * n*(4*n + 5).
(End)

A158620 Partial products of A068601.

Original entry on oeis.org

7, 182, 11466, 1421784, 305683560, 104543777520, 53421870312720, 38891121587660160, 38852230466072499840, 51673466519876424787200, 89240076679826585607494400, 195971208388899181994057702400

Offset: 2

Jonathan Vos Post, Mar 23 2009

A158621(n) = Product_{k=2..n} (k^3+1). A158622(n) is the numerator of the reduced fraction A158620(n)/A158621(n). A158623(n) is the denominator of the reduced fraction A158620(n)/A158621(n).

Also the determinant of the n X n matrix given by m(i,j) = i^3 if i=j and 1 otherwise. For example, Det[{{1,1,1, 1},{1,8,1,1},{1,1,27,1},{1,1,1,64}}] = 11466 = a(4). - John M. Campbell, May 20 2011

			a(2) = 2^3-1 = 7.
a(3) = (2^3-1)*(3^3-1) = 7 * 26 = 182.
a(4) = (2^3-1)*(3^3-1)*(4^3-1) = 7 * 26 * 63 = 11466.

G. C. Greubel, Table of n, a(n) for n = 2..180

Cf. A000217, A005448, A016921, A068601, A158621-A158624, A010791.

Rest[FoldList[Times,1,Range[2,15]^3-1]] (* Harvey P. Dale, Apr 18 2015 *)

a(n) = prod(k = 2, n, k^3 - 1); \\ Michel Marcus, Sep 29 2013

Product_{k=2..n} (k^3-1) = Product_{k=2..n} A068601(k).

a(n) ~ 2^(3/2) * sqrt(Pi) * cosh(sqrt(3)*Pi/2) * n^(3*n+3/2) / (3 * exp(3*n)). - Vaclav Kotesovec, Jul 11 2015

A303000 a(n) = permanent of the n X n matrix with entries a(i, i) = i^2 and a(i, j) = 1 elsewhere.

Original entry on oeis.org

1, 5, 52, 918, 24630, 934938, 47736048, 3157054776, 262661665176, 26857133054424, 3311299323349920, 484541686800059760, 83031688670103506160, 16472545369548670950480, 3746065113561656467249920, 968109978211279792380074880, 282158259444905145777416119680

Offset: 1

Vaclav Kotesovec, Apr 17 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..36

Cf. A007840, A010791, A067550, A118714, A303001.

Table[Permanent[DiagonalMatrix[Table[i^2-1, {i, 1, n}]] + 1], {n, 1, 20}]

{a(n) = matpermanent(matrix(n, n, i, j, if(i==j, i^2, 1)))}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 20 2018

A303001 a(n) = permanent of the n X n matrix with entries a(i, i) = i*(i+1)/2 and a(i, j) = 1 elsewhere.

Original entry on oeis.org

1, 4, 30, 356, 6106, 142760, 4363848, 168986136, 8087082144, 468807362736, 32379640476000, 2627735592279600, 247610398718738640, 26815386224063189760, 3307855985755600598400, 461149884030679844958720, 72151124506962747825006720, 12590610689213961622942752000

Offset: 1

Vaclav Kotesovec, Apr 17 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..36

Cf. A007840, A010791, A067550, A118714, A303000.

Table[Permanent[DiagonalMatrix[Table[i*(i+1)/2-1, {i, 1, n}]] + 1], {n, 1, 20}]

{a(n) = matpermanent(matrix(n, n, i, j, if(i==j, i*(i+1)/2, 1)))}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 20 2018

A064633 a(n) = 3^nn!(n+2)!/2!.

Original entry on oeis.org

1, 9, 216, 9720, 699840, 73483200, 10581580800, 1999918771200, 479980505088000, 142554210011136000, 51319515604008960000, 22016072194119843840000, 11096100385836401295360000, 6491218725714294757785600000, 4362098983680006077231923200000

Offset: 0

Karol A. Penson, Oct 01 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 0..75

Cf. A000244, A010791.

Table[3^n n! (n+2)!/2,{n,0,20}] (* Harvey P. Dale, Feb 25 2015 *)

{ for (n=0, 75, write("b064633.txt", n, " ", 3^n*n!*(n + 2)!/2) ) } \\ Harry J. Smith, Sep 20 2009

Hypergeometric g.f.: (1-3*x)^(-3).
a(0)=1, a(n) = n!*subs(x=0, (d^n/dx^n)(-1/((3*x-1)^3))), n = 1, 2, ...
From Amiram Eldar, Sep 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 6*BesselI(2,2/sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 6*BesselJ(2,2/sqrt(3)). (End)

A319900 a(n) is the number of distinct ways to arrange n copies of each of the numbers 1 through n^2 inside a fixed n X n X n cube, provided that no number appears twice in the same left-right plane, front-back plane, or top-bottom plane.

Original entry on oeis.org

1, 24, 14515200, 7708721243457872461824000

Offset: 1

Tanya Khovanova and Wayne Zhao, Sep 30 2018

When n = 3, this is equivalent to enumerating the different fill-ins of a Sudo-Kurve puzzle of the shape given in the link 'Sudo-Kurve 38'.

			For n = 2, the top layer of the 2 X 2 X 2 cube must contain each of the numbers 1, 2, 3, 4. This can be arranged in 24 ways. Each way uniquely determines the rest of the cube, so there are 24 possible cubes.

Bert Dobbelaere, C++ program for a(4)
T. Khovanova and W. Zhao, Mathematics of a Sudo-Kurve, arXiv:1808.06713 [math.HO], 2018.
The Art of Puzzles, Sudo-Kurve 38

Cf. A107739, A109741.

Observation: a(n) = A010791(n*(n-1)) for 1 <= n <= 3. - Omar E. Pol, Oct 02 2018

a(4) from Bert Dobbelaere, Sep 20 2019

cp's OEIS Frontend

A302913 Determinant of n X n matrix whose main diagonal consists of the first n 9-gonal numbers and all other elements are 1's.

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A302914 Determinant of n X n matrix whose main diagonal consists of the first n 10-gonal numbers and all other elements are 1's.

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A158620 Partial products of A068601.

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A303000 a(n) = permanent of the n X n matrix with entries a(i, i) = i^2 and a(i, j) = 1 elsewhere.

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A303001 a(n) = permanent of the n X n matrix with entries a(i, i) = i*(i+1)/2 and a(i, j) = 1 elsewhere.

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A064633 a(n) = 3^n*n!*(n+2)!/2!.

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A319900 a(n) is the number of distinct ways to arrange n copies of each of the numbers 1 through n^2 inside a fixed n X n X n cube, provided that no number appears twice in the same left-right plane, front-back plane, or top-bottom plane.

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A064633 a(n) = 3^nn!(n+2)!/2!.