A069778 -id:A069778 - cp's OEIS frontend

A251780 Digital root of A069778(n-1) = n^3 - n^2 + 1, n >= 1. Repeat(1, 6, 3, 7, 6, 6, 4, 6, 9).

1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9, 1, 6, 3, 7, 6, 6, 4, 6, 9

Offset: 1

Peter M. Chema, Dec 08 2014

Periodic with cycle of 9: {1, 6, 3, 7, 6, 6, 4, 6, 9}.

The decimal expansion of 54588823/333333333 = 0.repeat(163766469).

			For a(3) = 3 because 3^3 - 3^2 + 3  = 27 - 9 + 3 = 21 with digit sum 3 which is also the digital root of 21.

Eric Weisstein's World of Mathematics, Digital Root.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A069778, A251754, A010888, A056992, A073636.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 6, 3, 7, 6, 6, 4, 6, 9},108] (* Ray Chandler, Jul 25 2016 *)

DR(n)=s=sumdigits(n);while(s>9,s=sumdigits(s));s
for(n=1,100,print1(DR(abs(n^2-n-n^3)),", ")) \\ Derek Orr, Dec 30 2014

a(n) = digital root of n^3 - n^2 + n.

More terms from Derek Orr, Dec 30 2014

Edited: name changed; formula, comment and example rewritten; digital root link added. - Wolfdieter Lang, Jan 05 2015

A001614 Connell sequence: 1 odd, 2 even, 3 odd, ...

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122

Offset: 1

N. J. A. Sloane

Next (2n-1) odd numbers alternating with next 2n even numbers. Squares (A000290(n)) occur at the A000217(n)-th entry. - Lekraj Beedassy, Aug 06 2004. - Comment corrected by Daniel Forgues, Jul 18 2009
a(t_n) = a(n(n+1)/2) = n^2 relates squares to triangular numbers. - Daniel Forgues
The natural numbers not included are A118011(n) = 4n - a(n) as n=1,2,3,... - Paul D. Hanna, Apr 10 2006
As a triangle with row sums = A069778 (1, 6, 21, 52, 105, ...): /Q 1;/Q 2, 4;/Q 5, 7, 9;/Q 10, 12, 14, 16;/Q ... . - Gary W. Adamson, Sep 01 2008
The triangle sums, see A180662 for their definitions, link the Connell sequence A001614 as a triangle with six sequences, see the crossrefs. - Johannes W. Meijer, May 20 2011
a(n) = A122797(n) + n - 1. - Reinhard Zumkeller, Feb 12 2012

			From _Omar E. Pol_, Aug 13 2013: (Start)
Written as a triangle the sequence begins:
   1;
   2,  4;
   5,  7,  9;
  10, 12, 14, 16;
  17, 19, 21, 23, 25;
  26, 28, 30, 32, 34, 36;
  37, 39, 41, 43, 45, 47, 49;
  50, 52, 54, 56, 58, 60, 62, 64;
  65, 67, 69, 71, 73, 75, 77, 79, 81;
  82, 84, 86, 88, 90, 92, 94, 96, 98, 100;
  ...
Right border gives A000290, n >= 1.
(End)

C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 276.
C. A. Pickover, The Mathematics of Oz, Chapter 39, Camb. Univ. Press UK 2002.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Ian Connell and Andrew Korsak, Problem E1382, Amer. Math. Monthly, 67 (1960), 380.
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Gary E. Stevens, A Connell-Like Sequence, J. Integer Sequences, Vol. 1, 1998, #98.1.4.
Eric Weisstein's World of Mathematics, Connell Sequence

Cf. A117384, A118011 (complement), A118012.
Cf. A069778. - Gary W. Adamson, Sep 01 2008
From Johannes W. Meijer, May 20 2011: (Start)
Triangle columns: A002522, A117950 (n>=1), A117951 (n>=2), A117619 (n>=3), A154533 (n>=5), A000290 (n>=1), A008865 (n>=2), A028347 (n>=3), A028878 (n>=1), A028884 (n>=2), A054569 [T(2*n,n)].
Triangle sums (see the comments): A069778 (Row1), A190716 (Row2), A058187 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Fi1, Fi2, Ze1 and Ze2), A000292 (Related to Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4), A190717 (Related to Ca1, Ca2, Ze3, Ze4), A190718 (Related to Gi1 and Gi2). (End)
Cf. A014132, A057211, A023531.

a001614 n = a001614_list !! (n-1)
a001614_list = f 0 0 a057211_list where
   f c z (x:xs) = z' : f x z' xs where z' = z + 1 + 0 ^ abs (x - c)
-- Reinhard Zumkeller, Dec 30 2011

[2*n-Round(Sqrt(2*n)): n in [1..80]]; // Vincenzo Librandi, Apr 17 2015

A001614:=proc(n): 2*n - floor((1+sqrt(8*n-7))/2) end: seq(A001614(n),n=1..67); # Johannes W. Meijer, May 20 2011

lst={};i=0;For[j=1, j<=4!, a=i+1;b=j;k=0;For[i=a, i<=9!, k++;AppendTo[lst, i];If[k>=b, Break[]];i=i+2];j++ ];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
row[n_] := 2*Range[n+1]+n^2-1; Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)

a(n)=2*n - round(sqrt(2*n)) \\ Charles R Greathouse IV, Apr 20 2015

from math import isqrt
def A001614(n): return (m:=n<<1)-(k:=isqrt(m))-int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

a(n) = 2*n - floor( (1+ sqrt(8*n-7))/2 ).
a(n) = A005843(n) - A002024(n). - Lekraj Beedassy, Aug 06 2004
a(n) = A118012(A118011(n)). A117384( a(n) ) = n; A117384( 4*n - a(n) ) = n. - Paul D. Hanna, Apr 10 2006
a(1) = 1; then a(n) = a(n-1)+1 if a(n-1) is a square, a(n) = a(n-1)+2 otherwise. For example, a(21)=36 is a square therefore a(22)=36+1=37 which is not a square so a(23)=37+2=39 ... - Benoit Cloitre, Feb 07 2007
T(n,k) = (n-1)^2 + 2*k - 1. - Omar E. Pol, Aug 13 2013
a(n)^2 = a(n*(n+1)/2). - Ivan N. Ianakiev, Aug 15 2013
Let the sequence be written in the form of the triangle in the EXAMPLE section below and let a(n) and a(n+1) belong to the same row of the triangle. Then a(n)*a(n+1) + 1 = a(A000217(A118011(n))) = A000290(A118011(n)). - Ivan N. Ianakiev, Aug 16 2013
a(n) = 2*n-round(sqrt(2*n)). - Gerald Hillier, Apr 15 2015
From Robert Israel, Apr 20 2015 (Start):
G.f.: 2*x/(1-x)^2 - (x/(1-x))*Sum_{n>=0} x^(n*(n+1)/2) = 2*x/(1-x)^2 - (Theta2(0,x^(1/2)))*x^(7/8)/(2*(1-x)) where Theta2 is a Jacobi theta function.
a(n) = 2*n-1 - Sum_{i=0..n-2} A023531(i).  (End)
a(n) = 3*n-A014132(n). - Chai Wah Wu, Oct 19 2024

More terms from Larry Reeves (larryr(AT)acm.org), Mar 16 2001

A069777 Array of q-factorial numbers n!_q, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 21, 4, 1, 1, 1, 120, 315, 52, 5, 1, 1, 1, 720, 9765, 2080, 105, 6, 1, 1, 1, 5040, 615195, 251680, 8925, 186, 7, 1, 1, 1, 40320, 78129765, 91611520, 3043425, 29016, 301, 8, 1, 1

Offset: 0

Franklin T. Adams-Watters, Apr 07 2002

			Square array begins:
    1,   1,    1,      1,       1,        1,         1, ...
    1,   1,    1,      1,       1,        1,         1, ...
    1,   2,    3,      4,       5,        6,         7, ...
    1,   6,   21,     52,     105,      186,       301, ...
    1,  24,  315,   2080,    8925,    29016,     77959, ...
    1, 120, 9765, 251680, 3043425, 22661496, 121226245, ...
    ...

Alois P. Heinz, Antidiagonals n = 0..55, flattened
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Index entries for sequences related to factorial numbers

Columns q=0..11 are A000012, A000142, A005329, A015001, A015002, A015004, A015005, A015006, A015007, A015008, A015009, A015011.
Rows n=3..5 are A069778, A069779, A218503.
Main diagonal gives A347611.
Cf. A008302, A156173.

A069777 := proc(n,k) local n1: mul(A104878(n1,k), n1=k..n-1) end: A104878 := proc(n,k): if k = 0 then 1 elif k=1 then n elif k>=2 then (k^(n-k+1)-1)/(k-1) fi: end: seq(seq(A069777(n,k), k=0..n), n=0..9); # Johannes W. Meijer, Aug 21 2011
nmax:=9: T(0,0):=1: for n from 1 to nmax do T(n,0):=1:  T(n,1):= (n-1)! od: for q from 2 to nmax do for n from 0 to nmax do T(n+q,q) := product((q^k - 1)/(q - 1), k= 1..n) od: od: for n from 0 to nmax do seq(T(n,k), k=0..n) od; seq(seq(T(n,k), k=0..n), n=0..nmax); # Johannes W. Meijer, Aug 21 2011
# alternative Maple program:
T:= proc(n, k) option remember; `if`(n<2, 1,
      T(n-1, k)*`if`(k=1, n, (k^n-1)/(k-1)))
    end:
seq(seq(T(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Sep 08 2021

(* Returns the rectangular array *) Table[Table[QFactorial[n, q], {q, 0, 6}], {n, 0, 6}] (* Geoffrey Critzer, May 21 2017 *)

T(n,q)=prod(k=1, n, ((q^k - 1) / (q - 1))) \\ Andrew Howroyd, Feb 19 2018

T(n,q) = Product_{k=1..n} (q^k - 1) / (q - 1).
T(n,k) = Product_{n1=k..n-1} A104878(n1,k). - Johannes W. Meijer, Aug 21 2011
T(n,k) = Sum_{i>=0} A008302(n,i)*k^i. - Geoffrey Critzer, Feb 26 2025

Name edited by Michel Marcus, Sep 08 2021

A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 9, 16, 25, 1, 5, 12, 22, 35, 51, 1, 6, 15, 28, 45, 66, 91, 1, 7, 18, 34, 55, 81, 112, 148, 1, 8, 21, 40, 65, 96, 133, 176, 225, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451

Offset: 0

Paul Barry, Mar 21 2003

			The array starts
  1  1  3 10 ...
  1  2  6 16 ...
  1  3  9 22 ...
  1  4 12 28 ...
The triangle starts
  1;
  1,  1;
  1,  2,  3;
  1,  3,  6, 10;
  1,  4,  9, 16, 25;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Polygonal Number

Rows include A060354, A064808, A006000, A006003, A002411.
Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.
Antidiagonals are composed of n-gonal numbers.

Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # G. C. Greubel, Aug 14 2019

[[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // G. C. Greubel, Oct 13 2018

Table[PolygonalNumber[n,i],{n,0,10},{i,n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* Harvey P. Dale, Aug 27 2016 *)

tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

[[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # G. C. Greubel, Aug 14 2019

Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.

T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - Michel Marcus, Jun 22 2015

A226449 a(n) = n(5n^2-8*n+5)/2.

Original entry on oeis.org

0, 1, 9, 39, 106, 225, 411, 679, 1044, 1521, 2125, 2871, 3774, 4849, 6111, 7575, 9256, 11169, 13329, 15751, 18450, 21441, 24739, 28359, 32316, 36625, 41301, 46359, 51814, 57681, 63975, 70711, 77904, 85569, 93721, 102375, 111546, 121249, 131499, 142311, 153700

Offset: 0

Bruno Berselli, Jun 07 2013

Sequences of the type b(m)+m*b(m-1), where b is a polygonal number:
A006003(n) = A000217(n) + n*A000217(n-1)      (b = triangular numbers);
A069778(n) = A000290(n+1) + (n+1)*A000290(n)  (b = square numbers);
A143690(n) = A000326(n+1) + (n+1)*A000326(n)  (b = pentagonal numbers);
A212133(n) = A000384(n) + n*A000384(n-1)      (b = hexagonal numbers);
a(n)       = A000566(n) + n*A000566(n-1)      (b = heptagonal numbers);
A226450(n) = A000567(n) + n*A000567(n-1)      (b = octagonal numbers);
A226451(n) = A001106(n) + n*A001106(n-1)      (b = nonagonal numbers);
A204674(n) = A001107(n+1) + (n+1)*A001107(n)  (b = decagonal numbers).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. (see the comment) A000566, A006003, A069778, A143690, A204674, A212133, A226450, A226451.

Magma
```
[n*(5*n^2-8*n+5)/2: n in [0..40]];
```

I:=[0,1,9,39]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (5 n^2 - 8 n + 5)/2, {n, 0, 40}]
CoefficientList[Series[x (1 + 5 x + 9 x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{4,-6,4,-1},{0,1,9,39},50] (* Harvey P. Dale, May 19 2017 *)

a(n)=n*(5*n^2-8*n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1+5*x+9*x^2)/(1-x)^4.

a(n) - a(-n) = A008531(n) for n>0.

A270867 a(n) = n^3 + 2n^2 + 4n + 1.

Original entry on oeis.org

1, 8, 25, 58, 113, 196, 313, 470, 673, 928, 1241, 1618, 2065, 2588, 3193, 3886, 4673, 5560, 6553, 7658, 8881, 10228, 11705, 13318, 15073, 16976, 19033, 21250, 23633, 26188, 28921, 31838, 34945, 38248, 41753, 45466, 49393, 53540, 57913, 62518, 67361, 72448

Offset: 0

Vincenzo Librandi, Apr 01 2016

Numbers of the type (m+1)^3 - (m-1)*m. Similar sequences are: A069778 with the closed form (m+1)^3 - m*(m+1), A152015 with (m+1)^3 - (m+1)*(m+2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015 (page 19, 4th row; page 21, 3rd row).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A069778, A016957, A152015, A270109.

Magma
```
[n^3+2*n^2+4*n+1: n in [0..50]];
```

A270867:=n->n^3+2*n^2+4*n+1: seq(A270867(n), n=0..100); # Wesley Ivan Hurt, Apr 01 2016

Table[n^3 + 2 n^2 + 4 n + 1, {n, 0, 40}]

x='x+O('x^99); Vec((1+4*x-x^2+2*x^3)/(1-x)^4) \\ Altug Alkan, Apr 01 2016

for i in range(0,100):print(i**3+2*i**2+4*i+1) # Soumil Mandal, Apr 02 2016

O.g.f.: (1 + 4*x - x^2 + 2*x^3)/(1 - x)^4.
E.g.f.: (1 + 7*x + 5*x^2 + x^3)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = -A270109(-n-1). - Bruno Berselli, Apr 01 2016
a(n+2) - 2*a(n+1) + a(n) = A016957(n+1). - Wesley Ivan Hurt, Apr 02 2016

A153257 a(n) = n^3 - (n+1)^2.

Original entry on oeis.org

-1, -3, -1, 11, 39, 89, 167, 279, 431, 629, 879, 1187, 1559, 2001, 2519, 3119, 3807, 4589, 5471, 6459, 7559, 8777, 10119, 11591, 13199, 14949, 16847, 18899, 21111, 23489, 26039, 28767, 31679, 34781, 38079, 41579, 45287, 49209, 53351, 57719, 62319, 67157, 72239

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A045991, A069778, A081437, A140719.

Table[n^3-(n+1)^2,{n,0,40}] (* Harvey P. Dale, Oct 05 2022 *)

my(x='x+O('x^43)); Vec((x^3+5*x^2+x-1)/(x-1)^4) \\ Elmo R. Oliveira, Aug 27 2025

From Elmo R. Oliveira, Aug 27 2025: (Start)
G.f.: (-1 + x + 5*x^2 + x^3)/(1 - x)^4.
E.g.f.: (-1 + x)*(1 + 3*x + x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Elmo R. Oliveira, Aug 27 2025

A093966 Array read by antidiagonals: number of {112,221}-avoiding words.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 6, 21, 16, 5, 1, 6, 33, 52, 25, 6, 1, 6, 33, 124, 105, 36, 7, 1, 6, 33, 196, 345, 186, 49, 8, 1, 6, 33, 196, 825, 786, 301, 64, 9, 1, 6, 33, 196, 1305, 2586, 1561, 456, 81, 10, 1, 6, 33, 196, 1305, 6186, 6601, 2808, 657, 100, 11

Offset: 1

Ralf Stephan, Apr 20 2004

A(n,k) is the number of n-long k-ary words that simultaneously avoid the patterns 112 and 221.

			Array, A(n, k), begins as:
  1,  1,   1,    1,    1,     1,     1 ... 1*A000012(k);
  2,  4,   6,    6,    6,     6,     6 ... 2*A158799(k-1);
  3,  9,  21,   33,   33,    33,    33 ... ;
  4, 16,  52,  124,  196,   196,   196 ... ;
  5, 25, 105,  345,  825,  1305,  1305 ... ;
  6, 36, 186,  786, 2586,  6186,  9786 ... ;
  7, 49, 301, 1561, 6601, 21721, 51961 ... ;
Antidiagonal triangle, T(n, k), begins as:
  1;
  1, 2;
  1, 4,  3;
  1, 6,  9,   4;
  1, 6, 21,  16,    5;
  1, 6, 33,  52,   25,    6;
  1, 6, 33, 124,  105,   36,    7;
  1, 6, 33, 196,  345,  186,   49,   8;
  1, 6, 33, 196,  825,  786,  301,  64,  9;
  1, 6, 33, 196, 1305, 2586, 1561, 456, 81, 10;

G. C. Greubel, Antidiagonals n = 1..50, flattened
A. Burstein and T. Mansour, Words restricted by patterns with at most 2 distinct letters, arXiv:math/0110056 [math.CO], 2001.

Cf. A069778, A093963 (antidiagonal sums), A093964, A093965 (main diagonal).

A[n_, k_]:= A[n, k]= If[n==1, 1, If[k==1, n, If[2<=kG. C. Greubel, Dec 29 2021 *)

A(n,k) = if(n >= k+1, sum(j=1, k, j*j!*binomial(k,j)), if(n<2, if(n<1, 0, k), n!*binomial(k,n) + sum(j=1, n-1, j*j!*binomial(k,j))));
T(n,k) = A(n-k+1, k);
for(n=1, 15, for(k=1, n, print1(T(n, k), ", ") ) )

@CachedFunction
def A(n,k):
    if (n==1): return 1
    elif (k==1): return n
    elif (2 <= k < n+1): return factorial(k)*binomial(n,k) + sum( j*factorial(j)*binomial(n,j) for j in (1..k-1) )
    else: return sum( j*factorial(j)*binomial(n,j) for j in (1..n) )
def T(n,k): return A(k, n-k+1)
flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Dec 29 2021

A(n, k) = k!*binomial(n, k) + Sum_{j=1..k-1} j*j!*binomial(n, j), for 2 <= k <= n, otherwise Sum_{j=1..n} j*j!*binomial(n, j), with A(1, k) = 1 and A(n, 1) = n.
From G. C. Greubel, Dec 29 2021: (Start)
T(n, k) = A(k, n-k+1).
Sum_{k=1..n} T(n, k) = A093963(n).
T(n, 1) = 1.
T(n, n) = n.
T(n, n-1) = (n-1)^2.
T(n, n-2) = A069778(n).
T(2*n-1, n) = A093965(n).
T(2*n, n) = A093964(n), for n >= 1. (End)

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.

cp's OEIS Frontend

A251780 Digital root of A069778(n-1) = n^3 - n^2 + 1, n >= 1. Repeat(1, 6, 3, 7, 6, 6, 4, 6, 9).

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A001614 Connell sequence: 1 odd, 2 even, 3 odd, ...

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A069777 Array of q-factorial numbers n!_q, read by ascending antidiagonals.

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A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

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A226449 a(n) = n*(5*n^2-8*n+5)/2.

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A270867 a(n) = n^3 + 2*n^2 + 4*n + 1.

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A226449 a(n) = n(5n^2-8*n+5)/2.

A270867 a(n) = n^3 + 2n^2 + 4n + 1.