A005329 -id:A005329 - cp's OEIS frontend

A275706 a(n) = Re([n]_{1+i}!), where [n]_q! is the q-factorial, i = sqrt(-1).

1, 1, 2, 1, -40, 135, -860, 20145, -137100, -6726225, -212460900, -3642898575, 654642826500, -26505894416625, 3335048243533500, -1368325090374591375, 133951676745003682500, 123266968248328746879375, 63057521158814641016317500, 17732380504905960076345280625

Offset: 0

Vladimir Reshetnikov, Sep 13 2016

sign

Robert Israel, Table of n, a(n) for n = 0..114
Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A276688 (imaginary part), A005329.

a:= n-> Re(mul(((1+I)^j-1)/((1+I)-1), j=1..n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 14 2016

Re@Table[QFactorial[n, 1 + I], {n, 0, 20}]

A075272 BinomialMean (BM) transform of A075271, which see for the definition of (BM).

Original entry on oeis.org

1, 2, 6, 34, 422, 11586, 678982, 82653026, 20565923814, 10362872458882, 10517568142605446, 21434335059927667362, 87558678536857464017446, 716228573446369122069676994, 11725371140175829761708518252742

Offset: 0

John W. Layman, Sep 11 2002

a(n) = 2*A075271(n-1), for n >= 1.

Binomial transform of A005329. - Vladimir Reshetnikov, Nov 20 2015

Alois P. Heinz, Table of n, a(n) for n = 0..50
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Cf. A075271, A005329.

iBM:= proc(p) proc (n) option remember; add (2^(k) *p(k) *(-1)^(n-k) *binomial(n,k), k=0..n) end end: a:='a': aa:= iBM(a): a:= n-> `if` (n=0, 1, 2*aa(n-1)): seq (a(n), n=0..16); # Alois P. Heinz, Sep 09 2008

Table[Sum[QFactorial[k, 2] Binomial[n, k], {k, 0, n}], {n, 0, 15}] (* Vladimir Reshetnikov, Oct 16 2016 *)

G.f.: Sum_{n>=0} x^n*Product_{i=1..n}(2^i/(1+(2^i-1)*x)). - Vladeta Jovovic, Mar 10 2008

O.g.f. as a continued fraction of Stieltjes's type: 1/(1 - 2*x/(1 - x/(1 - 2^3*x/(1 - 3^2*x/(1 - 2^5*x/(1 - 7^2*x/(1 - 2^7*x/(1 - 15^2*x/(1 - 2^9*x/(1 - 31^2*x - ... )))))))))). Cf. A005329. - Peter Bala, Nov 10 2017

More terms from Alois P. Heinz, Sep 09 2008

A152552 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=2.

Original entry on oeis.org

1, 1, 7, 148, 7611, 872341, 213651052, 109327540680, 115381584785027, 249159124679346991, 1095244903267253760231, 9765839519517673327876328, 176188639876138769279299798900, 6419535615261099235478072782943388

Offset: 0

Paul D. Hanna, Dec 07 2008

nonn

			G.f.: A(x) = 1 + x + 7/3*x^2 + 148/21*x^3 + 7611/315*x^4 + 872341/9765*x^5 +...
e_q(x,2) = 1 + x + x^2/3 + x^3/21 + x^4/315 + x^5/9765 + x^6/615195 +...
The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1).

Cf. A152550, A152551 (q=-1), A152553 (q=3); A005329.

{a(n,q=2)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=sqrt(serreverse(x/(e_q+x*O(x^n))^2)/x)); polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))}

G.f. satisfies: A(x) = e_q( x*A(x)^2, 2) and A( x/e_q(x,2)^2 ) = e_q(x,2) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
G.f.: A(x) = Sum_{n>=0} a(n)*x^n/faq(n,2) where faq(n,2) = q-factorial of n at q=2.
G.f.: A(x) = [(1/x)*Series_Reversion( x/e_q(x,2)^2 )]^(1/2)
a(n) = Sum_{k=0..n(n-1)/2} A152550(n,k)*2^k.

A204243 Determinant of the n-th principal submatrix of A204242.

Original entry on oeis.org

1, 2, 11, 144, 4149, 251622, 31340799, 7913773980, 4024015413705, 4106387069191890, 8395359475529822355, 34357677843892688699400, 281336437060919094044274525, 4608419756389534634440592965950, 150992374805715685629827976712244775

Offset: 1

Clark Kimberling, Jan 13 2012

nonn

Colin Barker, Table of n, a(n) for n = 1..80
Wikipedia, Matrix determinant lemma

Cf. A005329, A203011, A204238, A204242.

f:= n -> (1 - add(1/(2^i-1),i=2..n))*mul(2^i-1,i=2..n):
seq(f(n),n=1..30); # Robert Israel, Nov 30 2015

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := 2^i - 1;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]     (* A204242 *)
Table[Det[m[n]], {n, 1, 15}]  (* A204243 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 15}]   (* A203011 *)

vector(20, n, matdet(matrix(n, n, i, j, if(i==1, 1, if(j==1, 1, if(i==j, 2^i-1)))))) \\ Colin Barker, Nov 27 2015

a(n) = (1 - Sum_{k=2..n} 1/(2^k-1)) * Product_{k=2..n} (2^k-1) = 2*A005329(n) - A203011(n). - Robert Israel, Nov 30 2015

A274983 a(n) = [n]phi! + [n]{1-phi}!, where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.

Original entry on oeis.org

2, 2, 3, 14, 130, 2120, 58120, 2636360, 196132320, 23805331920, 4698862837680, 1505416321070640, 781888977967152000, 657866357975539785600, 896265744457831561756800, 1976607903479486428467148800, 7055269158071576119808840371200

Offset: 0

Vladimir Reshetnikov, Sep 23 2016

nonn

			For n = 3, [3]_phi! = 1060 + 474*sqrt(5), so a(5) = 2*1060 = 2120 and A274985(5) = 2*474 = 948.

Eric Weisstein's World of Mathematics, q-Factorial, Golden Ratio.

Cf. A274985, A005329, A275706, A276474, A276688, A276987.

Round@Table[QFactorial[n, GoldenRatio] + QFactorial[n, 1 - GoldenRatio], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

a(n) ~ c * phi^(n*(n+3)/2), where c = QPochhammer(phi-1) = A276987 = 0.1208019218617061294237231569887920563043992516794... . - Vaclav Kotesovec, Sep 24 2016
From Vladimir Reshetnikov, Sep 24 2016 (Start)
[n]_phi! = (a(n) + A274985(n)*sqrt(5))/2.
[n]_{1-phi}! = (a(n) - A274985(n)*sqrt(5))/2. (End)

A276688 a(n) = Im([n]_{1+i}!), where [n]_q! is the q-factorial, i = sqrt(-1).

Original entry on oeis.org

0, 0, 1, 8, 5, -220, 1895, -9140, -302175, -2778300, -95631825, -10071428100, -236788407375, 57706241794500, -7412904844112625, 525300693117661500, 348922898045520800625, 55166584329677385922500, 28368558145043150339199375, 46873210124734003815040957500

Offset: 0

Vladimir Reshetnikov, Sep 13 2016

sign

Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A275706 (real part), A005329.

a:= n-> Im(mul(((1+I)^j-1)/((1+I)-1), j=1..n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 14 2016

Im@Table[QFactorial[n, 1 + I], {n, 0, 20}]

A028692 23-factorial numbers.

Original entry on oeis.org

1, 22, 11616, 141320256, 39547060439040, 254538406080331591680, 37680818974206486508802211840, 128296611269497862923425473853914480640, 10047034036599529256387830050150921763777884979200, 18096242094820543236399273859296273669601076798103392511590400

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..38
Index entries for sequences related to factorial numbers.

Cf. A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880, A028693, A028694.

FoldList[ #1 (23^#2-1)&, 1, Range[ 20 ] ]
a[n_] := Abs[QPochhammer[23, 23, n]]; Array[a, 10, 0] (* Amiram Eldar, Jul 14 2025 *)

a(n) = prod(k = 1, n, 23^k - 1); \\ Amiram Eldar, Jul 14 2025

From Amiram Eldar, Jul 14 2025: (Start)
a(n) = Product_{k=1..n} (23^k-1).
a(n) ~ c * 23^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/23^k) = 0.954631535623... . (End)

A028693 24-factorial numbers.

Original entry on oeis.org

1, 23, 13225, 182809175, 60651514035625, 482945140644890444375, 92292253139031982469134515625, 423295781586452233477722435457009484375, 46594416147080909523690749946376478698532878515625, 123093479909646650570543074660375014342475500150254964721484375

Offset: 1

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..38
Index entries for sequences related to factorial numbers.

Cf. A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880, A028692, A028694.

FoldList[ #1 (24^#2-1)&, 1, Range[ 20 ] ]
a[n_] := Abs[QPochhammer[24, 24, n]]; Array[a, 10, 0] (* Amiram Eldar, Jul 14 2025 *)

a(n) = prod(k = 1, n, 24^k - 1); \\ Amiram Eldar, Jul 14 2025

From Amiram Eldar, Jul 14 2025: (Start)
a(n) = Product_{k=1..n} (24^k-1).
a(n) ~ c * 24^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/24^k) = 0.956597348026... . (End)

A028694 25-factorial numbers.

Original entry on oeis.org

1, 24, 14976, 233985024, 91400166014976, 892579654839833985024, 217914953902301689160166014976, 1330047325845938129350664710839833985024, 202949115880923695556030391039325175289160166014976, 774189437411767935420978172981557217629743778824710839833985024

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..37
Index entries for sequences related to factorial numbers.

Cf. A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880, A028692, A028693.

FoldList[ #1 (25^#2-1)&, 1, Range[ 20 ] ]
a[n_] := Abs[QPochhammer[25, 25, n]]; Array[a, 10, 0] (* Amiram Eldar, Jul 14 2025 *)

a(n) = prod(k = 1, n, 25^k - 1); \\ Amiram Eldar, Jul 14 2025

From Amiram Eldar, Jul 14 2025: (Start)
a(n) = Product_{k=1..n} (25^k-1).
a(n) ~ c * 25^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/25^k) = 0.958400102563... . (End)

A157638 Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 2.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 60, 210, 60, 1, 1, 155, 1550, 1550, 155, 1, 1, 378, 9765, 27900, 9765, 378, 1, 1, 889, 56007, 413385, 413385, 56007, 889, 1, 1, 2040, 302260, 5440680, 14055090, 5440680, 302260, 2040, 1, 1, 4599, 1563660, 66194940

Offset: 0

Roger L. Bagula, Mar 03 2009

Other triangles in the family (see name) include: q = 2 (this triangle), q = 3 (see A157640), and q = 4 (see A157641). - Werner Schulte, Nov 16 2018

			Triangle begins:
  1;
  1, 1;
  1, 6, 1;
  1, 21, 21, 1;
  1, 60, 210, 60, 1;
  1, 155, 1550, 1550, 155, 1;
  1, 378, 9765, 27900, 9765, 378, 1;
  1, 889, 56007, 413385, 413385, 56007, 889, 1;
  1, 2040, 302260, 5440680, 14055090, 5440680, 302260, 2040, 1;
  1, 4599, 1563660, 66194940, 417028122, 417028122, 66194940, 1563660, 4599, 1;

Andrew Howroyd, Rows n=0..49 of triangle, flattened

Cf. A007318, A005329, A022166, A157640, A157641.

q:=2; [[k le 0 select 1 else Binomial(n,k)*(&*[(1-q^(n-j))/(1-q^(j+1)): j in [0..(k-1)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 17 2018

t[n_, m_] = Product[Sum[k*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}];
b[n_, k_, m_] = t[n, m]/(t[k, m]*t[n - k, m]);
Flatten[Table[Table[b[n, k, 1], {k, 0, n}], {n, 0, 10}]]

T(n,k) = {binomial(n,k)*polcoef(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)} \\ Andrew Howroyd, Nov 14 2018

q=2; for(n=0,10, for(k=0,n, print1(binomial(n,k)*prod(j=0,k-1, (1-q^(n-j))/(1-q^(j+1))), ", "))) \\ G. C. Greubel, Nov 17 2018

[[ binomial(n,k)*gaussian_binomial(n,k).subs(q=2) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Nov 17 2018

T(n,k) = t(n)/(t(k)*t(n-k)) where t(n) = Product_{k=1..n} Sum_{i=0..k-1} k*2^i.
T(n,k) = binomial(n,k)*A022166(n,k) for 0 <= k <= n. - Werner Schulte, Nov 14 2018
T(n,k) = n!*A005329(n)/(k!*A005329(k)*(n-k)!*A005329(n-k)). - Andrew Howroyd, Nov 14 2018

Edited and simpler name by Werner Schulte and Andrew Howroyd, Nov 14 2018

cp's OEIS Frontend

A275706 a(n) = Re([n]_{1+i}!), where [n]_q! is the q-factorial, i = sqrt(-1).

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A075272 BinomialMean (BM) transform of A075271, which see for the definition of (BM).

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A152552 Coefficients in a q-analog of the function [LambertW(-2x)/(-2x)]^(1/2) at q=2.

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PARI

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A204243 Determinant of the n-th principal submatrix of A204242.

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A274983 a(n) = [n]phi! + [n]{1-phi}!, where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.

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A276688 a(n) = Im([n]_{1+i}!), where [n]_q! is the q-factorial, i = sqrt(-1).

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Maple

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A028692 23-factorial numbers.

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Mathematica

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A028693 24-factorial numbers.

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PARI