A007526 -id:A007526 - cp's OEIS frontend

A026243 a(n) = A000522(n) - 2.

0, 3, 14, 63, 324, 1955, 13698, 109599, 986408, 9864099, 108505110, 1302061343, 16926797484, 236975164803, 3554627472074, 56874039553215, 966858672404688, 17403456103284419, 330665665962403998, 6613313319248079999, 138879579704209680020, 3055350753492612960483

Offset: 1

N. J. A. Sloane, based on a message from a correspondent who wishes to remain anonymous, Dec 21 2003

nonn

Number of operations of addition and multiplication needed to evaluate a determinant of order n by cofactor expansion.

			To calculate a determinant of order 3:
    |a b c|       |e f|       |d f|       |d e|
D = |d e f| = a * |h i| - b * |g i| + c * |g h| =
    |g h i|
   = a * (e*i - f*h) - b * (d*i - f*g) + c * (d*h - e*g).
There are 9 multiplications * and 5 additions (+ or -), so 14 operations and a(3) = 14. - _Bernard Schott_, Apr 21 2019

Alois P. Heinz, Table of n, a(n) for n = 1..170
C. Dubbs and D. Siegel, Computing determinants, College Math. J., 18 (1987), 48-49.
A. R. Pargeter, The vanishing coffee morning, Math. Gaz., 76 (1992), 386-387.
P. G. Sawtelle, The ubiquitous e, Math. Mag., 49 (1976), 244-245. [_N. J. A. Sloane_, Jan 29 2009]

Cf. A000522, A007526. Equals A033312 + A038156.

Cf. A001339.

a:= proc(n) a(n):= n*(a(n-1)+2)-1: end: a(1):= 0:
seq (a(n), n=1..30);  # Alois P. Heinz, May 25 2012

Table[E*Gamma[n+1, 1] - 2, {n, 1, 30}] (* Jean-François Alcover, May 18 2018 *)

a(n) = n*(a(n-1)+2)-1 for n>1, a(1) = 0. - Alois P. Heinz, May 25 2012
Conjecture: a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Jun 23 2013 [Confirmed by Altug Alkan, May 18 2018]
a(n) = floor(e*n!) - 2. - Bernard Schott, Apr 21 2019

A073591 a(n) = A000522(n) + 1.

Original entry on oeis.org

2, 3, 6, 17, 66, 327, 1958, 13701, 109602, 986411, 9864102, 108505113, 1302061346, 16926797487, 236975164806, 3554627472077, 56874039553218, 966858672404691, 17403456103284422, 330665665962404001, 6613313319248080002

Offset: 0

Vladeta Jovovic, Aug 28 2002

nonn

a(n) is an upper bound on the Ramsey numbers in A003323. - D. G. Rogers, Aug 27 2006

There is a nice derivation of the recurrence relation given in the Walker reference.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (28 terms from Vincenzo Librandi)
R. C. Walker, A graph coloring theorem, Math. Gaz., 60 (1976), 54-57.

Cf. A000522, A001339, A003323, A007526.

a:= proc(n) a(n):= `if`(n=0, 2, n*a(n-1)-n+2) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 17 2014

f[n_] := n*(f[n - 1] - 1) + 2;f[0]=2; ff[n_]:=(1/(1+n))(1+E*Gamma[1+n, 1]-E*(n^2)*Gamma[1+n, 1]+E*n*Gamma[2+n, 1]) (Spindler)
Table[FunctionExpand[Gamma[n, 1] E] + 1, {n, 2, 29}] (* Vincenzo Librandi, Feb 17 2014 *)

Conjecture: a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Feb 16 2014

a(n) = n*(a(n-1) - 1) + 2. - Georg Fischer, Dec 24 2023 [from the Walker reference, p. 55]

A121662 Triangle read by rows: T(i,j) for the recurrence T(i,j) = (T(i-1,j) + 1)*i.

Original entry on oeis.org

1, 4, 2, 15, 9, 3, 64, 40, 16, 4, 325, 205, 85, 25, 5, 1956, 1236, 516, 156, 36, 6, 13699, 8659, 3619, 1099, 259, 49, 7, 109600, 69280, 28960, 8800, 2080, 400, 64, 8, 986409, 623529, 260649, 79209, 18729, 3609, 585, 81, 9, 9864100, 6235300, 2606500, 792100, 187300, 36100, 5860, 820, 100, 10

Offset: 1

Thomas Wieder, Aug 15 2006

The first column is A007526 = "the number of (nonnull) "variations" of n distinct objects, namely the number of permutations of nonempty subsets of {1,...,n}." E.g. for n=3 there are 15 subsets: {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}. These are subsets with a number of elements l=1,...,n. The second column excludes all subsets with l=n elements. For n=3 one has therefore only the 9 subsets {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}. The third column excludes all subsets with l>=n-1 elements. For n=3 one has therefore only the 3 subsets {a}, {b},{c}. See also A121684. The second column is A038156 = n!*Sum(1/k!, k=1..n-1). The first lower diagonal are the squares A000290 = n^2. The second lower diagonal (15, 40, 85...) is A053698 = n^3 + n^2 + n + 1. The row sum is A030297 = a(n) = n*(n+a(n-1)).

T(i, j) is the total number of ordered sets of size 1 to i-j+1 that can be created from i distinct items. - Manfred Boergens, Jun 22 2022

			Triangle T(i,j) begins:
       1
       4     2
      15     9     3
      64    40    16     4
     325   205    85    25    5
    1956  1236   516   156   36   6
   13699  8659  3619  1099  259  49  7
   ...

Cf. A007526, A030297, A053698, A038156, A000290, A121682.

Mirror of triangle A285268.

T:= proc(i, j) option remember;
      `if`(j<1 or j>i, 0, T(i-1, j)*i+i)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 22 2022

Table[Sum[m!/(m - i)!, {i, n}], {m, 9}, {n, m, 1, -1}] // Flatten (* Michael De Vlieger, Apr 22 2017 *)
(* Sum-free code *)
b[j_] = If[j == 0, 0, Floor[j! E - 1]];
T[i_, j_] = b[i] - i! b[j - 1]/(j - 1)!;
Table[T[i, j], {i, 24}, {j, i}] // Flatten
(* Manfred Boergens, Jun 22 2022 *)

From Manfred Boergens, Jun 22 2022: (Start)
T(i, j) = Sum_{k=1..i-j+1} i!/(i-k)! = Sum_{k=j-1..i-1} i!/k!.
Sum-free formula: T(i, j) = b(i) - i!*b(j-1)/(j-1)! where b(0)=0, b(j)=floor(j!*e-1) for j>0.
(End)

Edited by N. J. A. Sloane, Sep 15 2006

Formula in name corrected by Alois P. Heinz, Jun 22 2022

A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 1, 4, 16, 0, 1, 6, 15, 65, 0, 1, 10, 27, 64, 326, 0, 1, 18, 57, 124, 325, 1957, 0, 1, 34, 135, 292, 645, 1956, 13700, 0, 1, 66, 345, 796, 1585, 3906, 13699, 109601, 0, 1, 130, 927, 2404, 4605, 9726, 27391, 109600, 986410, 0, 1, 258, 2577, 7804, 15145, 28926, 68425, 219192, 986409, 9864101

Offset: 0

Seiichi Manyama, Aug 14 2020

			Square array begins:
     1,    0,    0,    0,     0,     0,      0, ...
     2,    1,    1,    1,     1,     1,      1, ...
     5,    4,    6,   10,    18,    34,     66, ...
    16,   15,   27,   57,   135,   345,    927, ...
    65,   64,  124,  292,   796,  2404,   7804, ...
   326,  325,  645, 1585,  4605, 15145,  54645, ...
  1957, 1956, 3906, 9726, 28926, 98646, 374526, ...

Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Columns k=0..5 give A000522, A007526, A030297, A337001, A337002, A368719.
Main diagonal gives A256016.
Cf. A368724.

T[n_, k_] := n! * Sum[If[j == k == 0, 1, j^k]/j!, {j, 0, n}]; Table[T[k, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)

T(0,k) = 0^k and T(n,k) = n^k + n * T(n-1,k) for n>0.

E.g.f. of column k: B_k(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024

A368574 a(n) = n! * Sum_{k=0..n} binomial(k+2,3) / k!.

Original entry on oeis.org

0, 1, 6, 28, 132, 695, 4226, 29666, 237448, 2137197, 21372190, 235094376, 2821132876, 36674727843, 513446190362, 7701692856110, 123227085698576, 2094860456876761, 37707488223782838, 716442276251875252, 14328845525037506580, 300905756025787639951, 6619926632567328080946

Offset: 0

Seiichi Manyama, Dec 31 2023

Cf. A007526, A103519, A368575, A368576.

Cf. A000292, A337001.

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 2, binomial(2, k)*x^k/(k+1)!)*exp(x)/(1-x))))

a(0) = 0; a(n) = n*a(n-1) + binomial(n+2,3).

E.g.f.: x * (1+x+x^2/6) * exp(x) / (1-x).

A368575 a(n) = n! * Sum_{k=0..n} binomial(k+3,4) / k!.

Original entry on oeis.org

0, 1, 7, 36, 179, 965, 5916, 41622, 333306, 3000249, 30003205, 330036256, 3960436437, 51485675501, 720799459394, 10811991893970, 172991870307396, 2940861795230577, 52935512314156371, 1005774733968978364, 20115494679379576135, 422425388266971109461

Offset: 0

Seiichi Manyama, Dec 31 2023

Cf. A007526, A103519, A368574, A368576.

Cf. A000332, A337002.

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 3, binomial(3, k)*x^k/(k+1)!)*exp(x)/(1-x))))

a(0) = 0; a(n) = n*a(n-1) + binomial(n+3,4).

E.g.f.: x * (1+3*x/2+x^2/2+x^3/24) * exp(x) / (1-x).

A067273 a(n) = n(a(n-1)2+1), a(0) = 0.

Original entry on oeis.org

0, 1, 6, 39, 316, 3165, 37986, 531811, 8508984, 153161721, 3063234430, 67391157471, 1617387779316, 42052082262229, 1177458303342426, 35323749100272795, 1130359971208729456, 38432239021096801521, 1383560604759484854774, 52575302980860424481431, 2103012119234416979257260, 88326509007845513128804941

Offset: 0

Reinhard Zumkeller, Feb 21 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..403

Cf. A007526.

a := n -> n*hypergeom([1,1-n],[],-2):
seq(simplify(a(n)), n=0..17); # Peter Luschny, May 09 2017

FoldList[2 #1*#2 + #2 &, 0, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
a[n_] := 2^(n-1)*Sqrt[E]*n*Gamma[n,1/2];
Table[a[n] // FullSimplify, {n,0,20}] (* Gerry Martens, Jun 28 2015 *)
nxt[{n_,a_}]:={n+1,(n+1)(2*a+1)}; NestList[nxt,{0,0},20][[;;,2]] (* Harvey P. Dale, Jun 26 2023 *)

E.g.f.: x*exp(x)/(1-2*x). a(n) = n!*Sum_{k=1..n} 2^(k-1)/(n-k)! = n*A010844(n-1). - Vladeta Jovovic, Feb 09 2003
a(n) ~ n! * exp(1/2) * 2^(n-1). - Vaclav Kotesovec, Oct 05 2013
a(n) = n*hypergeom([1,1-n], [], -2). - Peter Luschny, May 09 2017
a(n) = Sum_{k=1..n} 2^(k-1)*k!*binomial(n,k). - Ridouane Oudra, Jun 15 2025

More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008

A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 18, 24, 1, 8, 36, 96, 120, 1, 10, 60, 240, 600, 720, 1, 12, 90, 480, 1800, 4320, 5040, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880, 1, 18, 216, 2016, 15120, 90720, 423360, 1451520

Offset: 0

Alford Arnold, Aug 19 2006

Row sums are 1,3,11,49,261,1631,... = A001339

a(n,k) = D(n+1,k+1) Array D in A253938 is part of a conjectured formula for F(n,p,r) that relates Dyck path peaks and returns. a(n,k) was discovered prior to array D. - Roger Ford, May 19 2016

			Row 6 is 1*1 5*2 10*6 10*24 5*120 1*720.
From _Vincenzo Librandi_, Dec 16 2012: (Start)
Triangle begins:
1,
1, 2,
1, 4,  6,
1, 6,  18,  24,
1, 8,  36,  96,   120,
1, 10, 60,  240,  600,  720,
1, 12, 90,  480,  1800, 4320,  5040,
1, 14, 126, 840,  4200, 15120, 35280,  40320,
1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880 etc.
(End)

Vincenzo Librandi, Rows n = 0..100, flattened
J. Goldman, J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients of k rooks on the 2xn board (all heights 2).
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007526 A000522, A005843 (2nd column), A028896 (3rd column).
Cf. A008279.
Cf. A008277, A132159 (mirrored).

a121757 n k = a121757_tabl !! n !! k
a121757_row n = a121757_tabl !! n
a121757_tabl = iterate
   (\xs -> zipWith (+) (xs ++ [0]) (zipWith (*) [1..] ([0] ++ xs))) [1]
-- Reinhard Zumkeller, Mar 06 2014

Flatten[Table[n!(k+1)/(n-k)!,{n,0,10},{k,0,n}]]  (* Harvey P. Dale, Apr 25 2011 *)

A000142(n)={ return(n!) ; } A007318(n,k)={ return(binomial(n,k)) ; } A121757(n,k)={ return(A007318(n,k)*A000142(k+1)) ; } { for(n=0,12, for(k=0,n, print1(A121757(n,k),",") ; ); ) ; } \\ R. J. Mathar, Sep 02 2006

a(n,k) = A007318(n,k)*A000142(k+1), k=0,1,..,n, n=0,1,2,3... - R. J. Mathar, Sep 02 2006

a(n,k) = A008279(n,k) * (k+1). a(n,k) = n!*(k+1)/(n-k)!. - Franklin T. Adams-Watters, Sep 20 2006

A126062 Array read by antidiagonals: see A128195 for details.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 15, 1, 1, 16, 65, 64, 1, 1, 25, 175, 511, 325, 1, 1, 36, 369, 2020, 4743, 1956, 1, 1, 49, 671, 5629, 27313, 52525, 13699, 1, 1, 64, 1105, 12736, 100045, 440896, 683657, 109600, 1, 1, 81, 1695, 25099, 280581, 2122449, 8390875, 10256775

Offset: 0

N. J. A. Sloane, Feb 28 2007

			Array begins:
[0]  1,  1,   1,     1,      1,       1,         1,          1,            1
[1]  1,  4,  15,    64,    325,    1956,     13699,     109600,       986409
[2]  1,  9,  65,   511,   4743,   52525,    683657,   10256775,    174369527
[3]  1, 16, 175,  2020,  27313,  440896,   8390875,  184647364,   4616348125
[4]  1, 25, 369,  5629, 100045, 2122449,  53163625, 1542220261,  50895431301
[5]  1, 36, 671, 12736, 280581, 7376356, 229151411, 8252263296, 338358810761

P. Luschny, Variants of Variations.

The second row counts the variations of n distinct objects A007526.

The second column is sequence A000290. The third column is sequence A005917.

A126062 := proc(k,n) if n = 0 then 1 ; else (n*k+1)*(A126062(k,n-1)+k^n) ; fi ; end: for diag from 0 to 10 do for k from diag to 0 by -1 do n := diag-k ; printf("%d, ",A126062(k,n)) ; od ; od ; # R. J. Mathar, May 18 2007

a[, 0] = 1; a[k, n_] := a[k, n] = (n*k+1)*(a[k, n-1]+k^n); Table[a[k-n, n], {k, 0, 10}, {n, 0, k}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

T(k, n) = (n*k+1)*(T(k, n-1) + k^n), T(k, 0) = 1. - Peter Luschny, Feb 26 2007

More terms from R. J. Mathar, May 18 2007

A180255 a(n) = n^2 * a(n-1) + n, a(0)=0.

Original entry on oeis.org

0, 1, 6, 57, 916, 22905, 824586, 40404721, 2585902152, 209458074321, 20945807432110, 2534442699285321, 364959748697086236, 61678197529807573897, 12088926715842284483826, 2720008511064514008860865, 696322178832515586268381456, 201237109682597004431562240801

Offset: 0

Groux Roland, Jan 17 2011

nonn

Integral_{x=0..1} x^n*BesselI(0,2*x^(1/2)) dx = A006040(n)*BesselI(1,2) - a(n)*BesselI(0,2). An elementary consequence is the irrationality of BesselI(0,2)/BesselI(1,2).

Paolo Xausa, Table of n, a(n) for n = 0..250 (terms 0..41 from Vincenzo Librandi)

Cf. A006040, A007526, A066998, A228229.

FoldList[#2^2*# + #2 &, Range[0, 20]] (* Paolo Xausa, Jun 19 2025 *)

a[0]:0$ a[n]:=n^2*a[n-1]+n$ makelist(a[n], n, 0, 15); /* Bruno Berselli, May 23 2011 */

a(n)=if(n==0,0,(n)^2*a(n-1)+(n));
for(n=0,12,print1(a(n),", "));  /* show terms */

From Seiichi Manyama, Jan 05 2024: (Start)
a(n) = (n!)^2 * Sum_{k=0..n} k/(k!)^2.
a(n) = n * A228229(n-1) for n > 0. (End)

cp's OEIS Frontend

A026243 a(n) = A000522(n) - 2.

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A073591 a(n) = A000522(n) + 1.

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A121662 Triangle read by rows: T(i,j) for the recurrence T(i,j) = (T(i-1,j) + 1)*i.

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A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

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A368574 a(n) = n! * Sum_{k=0..n} binomial(k+2,3) / k!.

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A368575 a(n) = n! * Sum_{k=0..n} binomial(k+3,4) / k!.

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A067273 a(n) = n*(a(n-1)*2+1), a(0) = 0.

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A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

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A067273 a(n) = n(a(n-1)2+1), a(0) = 0.