A069777 -id:A069777 - cp's OEIS frontend

A015007 q-factorial numbers for q=8.

1, 1, 9, 657, 384345, 1799118945, 67375205371305, 20185139902805378865, 48378633136349277767794425, 927610024989668734297857360967425, 142287668466497494704440569679875994730825, 174605966461872393482359052970987514818406771638225

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40
Index entries for sequences related to factorial numbers.

Cf. A015001, A015002, A015004, A015005, A015006, A015008, A015009, A015011.
Column q=8 of A069777.
Cf. A023001, A132036.

[n le 1 select 1 else (8^n-1)*Self(n-1)/7: n in [1..15]]; // Vincenzo Librandi, Oct 26 2012

RecurrenceTable[{a[1]==1, a[n]==((8^n - 1) * a[n-1])/7}, a, {n, 15}] (* Vincenzo Librandi, Oct 26 2012 *)
Table[QFactorial[n, 8], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} ((q^k - 1) / (q - 1)), with q=8.
a(0) = 1, a(n) = (8^n-1)*a(n-1)/7. - Vincenzo Librandi, Oct 26 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A023001(k).
a(n) ~ c * 8^(n*(n+1)/2)/7^n, where c = A132036. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A015008 q-factorial numbers for q=9.

Original entry on oeis.org

1, 1, 10, 910, 746200, 5507702200, 365876657146000, 218747042884536166000, 1177042838234827583459440000, 57001313848230245122464621625840000, 24843911488189148287648216529610193612000000, 97453533413342456299179976631323547842824103012000000

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40
Index entries for sequences related to factorial numbers.

Cf. A015001, A015002, A015004, A015005, A015006, A015007, A015009, A015011.
Column q=9 of A069777.
Cf. A002452, A132037.

[n le 1 select 1 else (9^n - 1)*Self(n-1)/8: n in [1..15]]; // Vincenzo Librandi, Oct 26 2012

RecurrenceTable[{a[1]==1, a[n]==((9^n - 1) * a[n-1])/8}, a, {n, 15}] (* Vincenzo Librandi, Oct 26 2012 *)
Table[QFactorial[n, 9], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} (9^k - 1) / (9 - 1).
a(0) = 1, a(n) = (9^n - 1)*a(n-1)/8. - Vincenzo Librandi, Oct 26 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A002452(k).
a(n) ~ c * 3^(n*(n+1))/8^n, where c = A132037. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A015009 q-factorial numbers for q=10.

Original entry on oeis.org

1, 1, 11, 1221, 1356531, 15072415941, 1674711207620451, 1860790044610366931061, 20675444733360738721748118771, 2297271634742810443154153338805764581, 2552524038347870310755413660544832496799359491, 28361378203581611893021499527080870668821235178133404501

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40
Index entries for sequences related to factorial numbers.

Cf. A015001, A015002, A015004, A015005, A015006, A015007, A015008, A015011.
Column q=10 of A069777.
Cf. A002275, A132038.

[n le 1 select 1 else (10^n-1)*Self(n-1)/9: n in [1..15]]; // Vincenzo Librandi, Oct 26 2012

RecurrenceTable[{a[1]==1, a[n]==((10^n-1) * a[n-1])/9}, a, {n, 15}] (* Vincenzo Librandi, Oct 26 2012 *)
Table[QFactorial[n, 10], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} (q^k - 1)/(q - 1) with q=10.
a(0) = 1, a(n) = (10^n - 1)*a(n-1)/9. - Vincenzo Librandi, Oct 26 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A002275(k).
a(n) ~ c * 10^(n*(n+1)/2)/9^n, where c = A132038. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A015011 q-factorial numbers for q=11.

Original entry on oeis.org

1, 1, 12, 1596, 2336544, 37630041120, 6666387564654720, 12990902775831251994240, 278471536921607824648305285120, 65662131721505488121539650946349537280, 170310659060181679663863033233125976844488908800, 4859161865915056755501262525796512204608930674134393036800

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40
Index entries for sequences related to factorial numbers.

Cf. A015001, A015002, A015004, A015005, A015006, A015007, A015008, A015009.
Column q=11 of A069777.
Cf. A016123, A132267.

[n le 1 select 1 else (11^n-1)*Self(n-1)/10: n in [1..15]]; // Vincenzo Librandi, Oct 26 2012

RecurrenceTable[{a[1]==1, a[n]==((11^n - 1) * a[n-1])/10}, a, {n, 15}] (* Vincenzo Librandi, Oct 26 2012 *)
Table[QFactorial[n, 11], {n, 11}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} (11^k - 1) / (11 - 1).
a(0) = 1, a(n) = (11^n - 1)*a(n-1)/10. - Vincenzo Librandi, Oct 26 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A016123(k-1).
a(n) ~ c * 11^(n*(n+1)/2)/10^n, where c = A132267. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A015005 q-factorial numbers for q=6.

Original entry on oeis.org

1, 1, 7, 301, 77959, 121226245, 1131162092095, 63330372050122765, 21274128570193389587095, 42878835824239014254983869205, 518543838148941095553869851505328175, 37625235473766496167083515195884075739704925, 16380389585902052954270520869620904155598347770499975

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Index entries for sequences related to factorial numbers.

Cf. A015001, A015002, A015004, A015006, A015007, A015008, A015009, A015011.
Column q=6 of A069777.
Cf. A003464, A132034.

[n le 1 select 1 else (6^n-1)*Self(n-1)/5: n in [1..15]]; // Vincenzo Librandi, Oct 25 2012

RecurrenceTable[{a[1]==1, a[n]==((6^n - 1) * a[n-1])/5}, a, {n, 15}] (* Vincenzo Librandi, Oct 25 2012 *)
Table[QFactorial[n, 6], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} (6^k-1)/(6-1).
a(0) = 1, a(n) = (6^n-1)*a(n-1)/5. - Vincenzo Librandi, Oct 25 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A003464(k).
a(n) ~ c * 6^(n*(n+1)/2)/5^n, where c = A132034. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A069779 q-factorial numbers 4!_q.

Original entry on oeis.org

1, 24, 315, 2080, 8925, 29016, 77959, 182400, 384345, 746200, 1356531, 2336544, 3847285, 6097560, 9352575, 13943296, 20276529, 28845720, 40242475, 55168800, 74450061, 99048664, 130078455, 168819840, 216735625, 275487576, 346953699, 433246240, 536730405, 660043800

Offset: 0

Franklin T. Adams-Watters, Apr 07 2002

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A069777-A069778, A218503.

Table[QFactorial[4, n], {n, 0, 29}] (* Arkadiusz Wesolowski, Nov 01 2012 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,24,315,2080,8925,29016,77959},30] (* Harvey P. Dale, Aug 30 2020 *)

a(n) = (n + 1)*(n^2 + n + 1)*(n^3 + n^2 + n + 1).

G.f.: (1 + 17*x + 8*x^2*(21 + 43*x) + 5*x^4*(35 + 3*x))/(1 - x)^7. - Arkadiusz Wesolowski, Nov 01 2012

A218503 q-factorial numbers 5!_q.

Original entry on oeis.org

1, 120, 9765, 251680, 3043425, 22661496, 121226245, 510902400, 1799118945, 5507702200, 15072415941, 37630041120, 87029433985, 188664603960, 386925380325, 756298318336, 1417430759745, 2559798038520, 4472991338725, 7589075296800, 12538953723681

Offset: 0

Arkadiusz Wesolowski, Oct 31 2012

Index entries for sequences related to factorial numbers
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A069777, A069778, A069779.

Table[QFactorial[5, n], {n, 0, 20}]
Join[{1},With[{f=Times@@Table[Total[n^Range[0,i]],{i,4}]},Table[f,{n,20}]]] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,120,9765,251680,3043425,22661496,121226245,510902400,1799118945,5507702200,15072415941},30] (* Harvey P. Dale, Sep 04 2017 *)

a(n) = (n + 1)*(n^2 + n + 1)*(n^3 + n^2 + n + 1)*(n^4 + n^3 + n^2 + n + 1).

G.f.: (1 + x*(109 + x*(8500 + x*(150700 + x*(792550 + x*(1454134 + x*(978436 + 5*x*(45788 + x*(3053 + 33*x)))))))))/(1 - x)^11.

A347611 a(n) is the n-th n-factorial number: a(n) = n!_n.

Original entry on oeis.org

1, 1, 3, 52, 8925, 22661496, 1131162092095, 1375009641495014400, 48378633136349277767794425, 57001313848230245122464621625840000, 2552524038347870310755413660544832496799359491, 4859161865915056755501262525796512204608930674134393036800

Offset: 0

Alois P. Heinz, Sep 08 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..36
Index entries for sequences related to factorial numbers

Main diagonal of A069777.

Cf. A366355.

b:= proc(n, k) option remember; `if`(n<2, 1,
      b(n-1, k)*(k^n-1)/(k-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..12);

Array[QFactorial[#, #] &, 12, 0] (* Michael De Vlieger, Sep 09 2021 *)

a(n) = if (n<=1, 1, prod(k=1, n, (n^k-1)/(n-1))); \\ Michel Marcus, Sep 09 2021

from math import prod
def a(n):
    return 1 if n <= 1 else prod((n**k - 1)//(n - 1) for k in range(1, n+1))
print([a(n) for n in range(12)]) # Michael S. Branicky, Sep 09 2021

a(n) = Product_{j=1..n} (n^j-1)/(n-1) for n > 1, a(0) = a(1) = 1.
a(n) = A069777(n,n).
a(n) ~ exp(1) * n^(n*(n-1)/2). - Vaclav Kotesovec, Jun 09 2025

A276823 a(n) = 3 * [3n]_2! / ([2n+1]_2! * [n+1]_2!), where [n]_q! is the q-factorial.

Original entry on oeis.org

1, 9, 1241, 2634489, 87807053113, 46414431022602681, 390913823614809035461305, 52571422826552549403006580802745, 113007269646365312407427675894837602068665, 3884802624238339577626451297006421856376970743148729

Offset: 1

Vladimir Reshetnikov, Sep 18 2016

nonn

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

Cf. A069777, A005329, A022166, A218449.

a:= n-> 3*mul((2^j-1), j=1..3*n)/
         (mul((2^j-1), j=1..2*n+1)*
          mul((2^j-1), j=1..n+1)):
seq(a(n), n=1..12);  # Alois P. Heinz, Sep 20 2016

Table[3 QFactorial[3 n, 2]/(QFactorial[2 n + 1, 2] QFactorial[n + 1, 2]), {n, 10}] (* or *)
Table[3 QBinomial[3 n, 2 n + 1, 2]/(1 - 3 * 2^n + 2^(2 n + 1)), {n, 10}]

a(n) ~ c * 2^((n-2)*(2*n+1)), where c = 3/QPochhammer(1/2, 1/2) = 3*A065446 = 3/A048651. - Vaclav Kotesovec, Sep 20 2016

A320354 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Product_{j=1..n} (k^j - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 16, 21, 0, 1, 4, 45, 416, 315, 0, 1, 5, 96, 2835, 33280, 9765, 0, 1, 6, 175, 11904, 722925, 8053760, 615195, 0, 1, 7, 288, 37625, 7428096, 739552275, 5863137280, 78129765, 0, 1, 8, 441, 98496, 48724375, 23205371904, 3028466566125, 12816818094080, 19923090075, 0

Offset: 0

Ilya Gutkovskiy, Oct 11 2018

			Square array begins:
  1,     1,        1,          1,            1,             1,  ...
  0,     1,        2,          3,            4,             5,  ...
  0,     3,       16,         45,           96,           175,  ...
  0,    21,      416,       2835,        11904,         37625,  ...
  0,   315,    33280,     722925,      7428096,      48724375,  ...
  0,  9765,  8053760,  739552275,  23205371904,  378832015625,  ...

Columns k=1..12 give A000007, A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880.

Cf. A069777, A225816.

Table[Function[k, Product[k^j - 1, {j, 1, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[k^(i (i + 1)/2) x^i/Product[(1 + k^j x), {j, 0, i}], {i, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten

G.f. of column k: Sum_{i>=0} k^(i*(i+1)/2)*x^i / Product_{j=0..i} (1 + k^j*x).

For asymptotics of column k see comment from Vaclav Kotesovec in A027880.

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A015007 q-factorial numbers for q=8.

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A015008 q-factorial numbers for q=9.

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A015009 q-factorial numbers for q=10.

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A015011 q-factorial numbers for q=11.

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A015005 q-factorial numbers for q=6.

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A069779 q-factorial numbers 4!_q.

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A218503 q-factorial numbers 5!_q.

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A347611 a(n) is the n-th n-factorial number: a(n) = n!_n.

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A276823 a(n) = 3 * [3n]_2! / ([2n+1]_2! * [n+1]_2!), where [n]_q! is the q-factorial.