A128195 -id:A128195 - cp's OEIS frontend

A126062 Array read by antidiagonals: see A128195 for details.

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

A005917 Rhombic dodecahedral numbers: a(n) = n^4 - (n - 1)^4.

Original entry on oeis.org

1, 15, 65, 175, 369, 671, 1105, 1695, 2465, 3439, 4641, 6095, 7825, 9855, 12209, 14911, 17985, 21455, 25345, 29679, 34481, 39775, 45585, 51935, 58849, 66351, 74465, 83215, 92625, 102719, 113521, 125055, 137345, 150415, 164289, 178991

Offset: 1

N. J. A. Sloane

Final digits of a(n), i.e., a(n) mod 10, are repeated periodically with period of length 5 {1,5,5,5,9}. There is a symmetry in this list since the sum of two numbers equally distant from the ends is equal to 10 = 1 + 9 = 5 + 5 = 2*5. Last two digits of a(n), i.e., a(n) mod 100, are repeated periodically with period of length 50. - Alexander Adamchuk, Aug 11 2006
a(n) = VarScheme(n,2) in the scheme displayed in A128195. - Peter Luschny, Feb 26 2007
If Y is a 3-subset of a 2n-set X then, for n >= 2, a(n-2) is the number of 4-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
The numbers are the constant number found in magic squares of order n, where n is an odd number, see the comment in A006003. A Magic Square of side 1 is 1; 3 is 15; 5 is 65 and so on. - David Quentin Dauthier, Nov 07 2008
Two times the area of the triangle with vertices at (0,0), ((n - 1)^2, n^2), and (n^2, (n - 1)^2). - J. M. Bergot, Jun 25 2013
Bisection of A006003. - Omar E. Pol, Sep 01 2018
Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), M(n,0) = 2*n^2 + 1 = A058331(n) and M(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n) begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals a(n+1). The first five rows of array M are [1, 7, 17, 31, 49, ...]; [3, 5, 15, 29, 47, ...]; [9, 11, 13, 27, 45, ...]; [19, 21, 23, 25, 43, ...]; [33, 35, 37, 39, 41, ...]. - J. M. Bergot, Jul 16 2013 [This contribution was moved here from A047926 by Petros Hadjicostas, Mar 08 2021.]
For n>=2, these are the primitive sides s of squares of type 2 described in A344332. - Bernard Schott, Jun 04 2021
(a(n) + 1) / 2 = A212133(n) is the number of cells in the n-th rhombic-dodecahedral polycube. - George Sicherman, Jan 21 2024

J. H. Conway and R. K. Guy, The Book of Numbers, p. 53.
E. Deza and M. M. Deza, Figurate Numbers, World Scientific Publishing, 2012, pp. 123-124.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Mario Defranco and Paul E. Gunnells, Hypergraph matrix models and generating functions, arXiv:2204.11361 [math.CO], 2022.
Milan Janjic, Two Enumerative Functions
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (9).
Andy Nicol, Illustration of Rhombic Dodecahedral Numbers
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Rhombic Dodecahedral Number.
Eric Weisstein's World of Mathematics, Nexus Number.
D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819-822.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A063493, A063494, A063495, A063496.
Column k=3 of A047969.
Cf. A128195, A176850, A005408, A176271, A212133.
Cf. A001844, A000583, A000290.
Cf. A007588, A000292, A000332, A002817, A342354.
Cf. A031215, A008292.
Cf. A016754, A344330, A344332.

a005917 n = a005917_list !! (n-1)
a005917_list = map sum $ f 1 [1, 3 ..] where
   f x ws = us : f (x + 2) vs where (us, vs) = splitAt x ws
-- Reinhard Zumkeller, Nov 13 2014

[n^4 - (n-1)^4: n in [1..50]]; // Vincenzo Librandi, Aug 01 2011

Table[n^4-(n-1)^4,{n,40}]  (* Harvey P. Dale, Apr 01 2011 *)
#[[2]]-#[[1]]&/@Partition[Range[0,40]^4,2,1] (* More efficient than the above Mathematica program because it only has to calculate each 4th power once *) (* Harvey P. Dale, Feb 07 2015 *)
Differences[Range[0,40]^4] (* Harvey P. Dale, Aug 11 2023 *)

a(n)=n^4-(n-1)^4 \\ Charles R Greathouse IV, Jul 31 2011

A005917_list, m = [], [24, -12, 2, 1]
for _ in range(10**2):
    A005917_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

a(n) = (2*n - 1)*(2*n^2 - 2*n + 1).
Sum_{i=1..n} a(i) = n^4 = A000583(n). First differences of A000583.
G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
More generally, g.f. for n^m - (n - 1)^m is Euler(m, x)/(1 - x)^m, where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292). E.g.f.: x*(exp(y/(1 - x)) - exp(x*y/(1 - x)))/(exp(x*y/(1 - x))-x*exp(y/(1 - x))). - Vladeta Jovovic, May 08 2002
a(n) = sum of the next (2*n - 1) odd numbers; i.e., group the odd numbers so that the n-th group contains (2*n - 1) elements like this: (1), (3, 5, 7), (9, 11, 13, 15, 17), (19, 21, 23, 25, 27, 29, 31), ... E.g., a(3) = 65 because 9 + 11 + 13 + 15 + 17 = 65. - Xavier Acloque, Oct 11 2003
a(n) = 2*n - 1 + 12*Sum_{i = 1..n} (i - 1)^2. - Xavier Acloque, Oct 16 2003
a(n) = (4*binomial(n,2) + 1)*sqrt(8*binomial(n,2) + 1). - Paul Barry, Mar 14 2004
Binomial transform of [1, 14, 36, 24, 0, 0, 0, ...], if the offset is 0. - Gary W. Adamson, Dec 20 2007
Sum_{i=1..n-1}(a(i) + a(i+1)) = 8*Sum_{i=1..n}(i^3 + i) = 16*A002817(n-1) for n > 1. - Bruno Berselli, Mar 04 2011
a(n+1) = a(n) + 2*(6*n^2 + 1) = a(n) + A005914(n). - Vincenzo Librandi, Mar 16 2011
a(n) = -a(-n+1). a(n) = (1/6)*(A181475(n) - A181475(n-2)). - Bruno Berselli, Sep 26 2011
a(n) = A045975(2*n-1,n) = A204558(2*n-1)/(2*n - 1). - Reinhard Zumkeller, Jan 18 2012
a(n+1) = Sum_{k=0..2*n+1} (A176850(n,k) - A176850(n-1,k))*(2*k + 1), n >= 1. - L. Edson Jeffery, Nov 02 2012
a(n) = A005408(n-1) * A001844(n-1) = (2*(n - 1) + 1) * (2*(n - 1)*n + 1) = A000290(n-1)*12 + 2 + a(n-1). - Bruce J. Nicholson, May 17 2017
a(n) = A007588(n) + A007588(n-1) = A000292(2n-1) + A000292(2n-2) + A000292(2n-3) = A002817(2n-1) - A002817(2n-2). - Bruce J. Nicholson, Oct 22 2017
a(n) = A005898(n-1) + 6*A000330(n-1) (cf. Deza, Deza, 2012, p. 123, Section 2.6.2). - Felix Fröhlich, Oct 01 2018
a(n) = A300758(n-1) + A005408(n-1). - Bruce J. Nicholson, Apr 23 2020
G.f.: polylog(-4, x)*(1-x)/x. See the Simon Plouffe formula above (with expanded numerator), and the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021

A007526 a(n) = n*(a(n-1) + 1), a(0) = 0.

Original entry on oeis.org

0, 1, 4, 15, 64, 325, 1956, 13699, 109600, 986409, 9864100, 108505111, 1302061344, 16926797485, 236975164804, 3554627472075, 56874039553216, 966858672404689, 17403456103284420, 330665665962403999, 6613313319248080000, 138879579704209680021, 3055350753492612960484

Offset: 0

N. J. A. Sloane and Robert G. Wilson v

Eighteenth- and nineteenth-century combinatorialists call this the number of (nonnull) "variations" of n distinct objects, namely the number of permutations of nonempty subsets of {1,...,n}. Some early references to this sequence are Izquierdo (1659), Caramuel de Lobkowitz (1670), Prestet (1675) and Bernoulli (1713). - Don Knuth, Oct 16 2001, Aug 16 2004
Stirling transform of A006252(n-1) = [0,1,1,2,4,14,38,...] is a(n-1) = [0,1,4,15,64,...]. - Michael Somos, Mar 04 2004
In particular, for n >= 1 a(n) is the number of nonempty sequences with n or fewer terms, each a distinct element of {1,...,n}. - Rick L. Shepherd, Jun 08 2005
a(n) = VarScheme(1,n). See A128195 for the definition of VarScheme(k,n). - Peter Luschny, Feb 26 2007
if s(n) is a sequence of the form s(0)=x, s(n)= n(s(n-1)+k), then s(n)= n!*x + a(n)*k. - Gary Detlefs, Jun 06 2010
Exponential convolution of factorials (A000142) and nonnegative integers (A001477). - Vladimir Reshetnikov, Oct 07 2016
For n > 0, a(n) is the number of maps f: {1,...,n} -> {1,...,n} satisfying equal(x,y) <= equal(f(x),f(y)) for all x,y, where equal(x,y) is n if x and y are equal and min(x,y) if not. Here equal(x,y) is the equality predicate in the n-valued Gödel logic, see e. g. the Wikipedia chapter on many-valued logics. - Mamuka Jibladze, Mar 12 2025

			G.f. = x + 4*x^2 + 15*x^3 + 64*x^4 + 325*x^5 + 1956*x^6 + 13699*x^7 + ...
Consider the nonempty subsets of the set {1,2,3,...,n} formed by the first n integers. E.g., for n = 3 we have {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. For each subset S we determine its number of parts, that is nprts(S). The sum over all subsets is written as sum_{S=subsets}. Then we have A007526 = Sum_{S=subsets} nprts(S)!. E.g., for n = 3 we have 1!+1!+1!+2!+2!+2!+3! = 15. - _Thomas Wieder_, Jun 17 2006
a(3)=15: Let the objects be a, b, and c. The fifteen nonempty ordered subsets are {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}, {abc}, {acb}, {bac}, {bca}, {cab} and {cba}.

Jacob Bernoulli, Ars Conjectandi (1713), page 127.
Johannes Caramuel de Lobkowitz, Mathesis Biceps Vetus et Nova (Campania: 1670), volume 2, 942-943.
J. K. Horn, personal communication to Robert G. Wilson v.
Sebastian Izquierdo, Pharus Scientiarum (Lyon: 1659), 327-328.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas, Freeman, San Francisco, 1974. [Annotated scans of pages 69 and 70 only]
J. Bernoulli, Wahrscheinlichkeitsrechnung (Ars conjectandi) von Jakob Bernoulli (1713) Uebers. und hrsg. von R. Haussner, Leipzig, W. Engelmann, (1899), [124] Kapitel VII. Variationen ohne Wiederholung. (Page 121).
Oscar Cabrera, Introducing loop compression for encoding de Bruijn sequences, engrXiv (2025) Art. No. 4431. See p. 16.
Peter J. Freyd, Core algebra revisited, Theoretical Computer Science, 375 (2007), Issues 1-3, 193-200.
Z. Kasa and Z. Katai, Scattered subwords and composition of natural numbers, Acta Univ. Sapientiae, Informatica, 4, 2 (2012) 225-236. - From _N. J. A. Sloane_, Feb 21 2013
Jean Prestet, Elemens des Mathematiques, (1675), page 341.
Joe Sawada and A. Williams, Successor rules for flipping pancakes and burnt pancakes, Preprint 2015.
Elmar Teufl and Stephan Wagner, Enumeration problems for classes of self-similar graphs, Journal of Combinatorial Theory, Series A, Volume 114, Issue 7, October 2007, Pages 1254-1277.

Row sums of A068424.
Partial sums of A001339.
Column k=1 of A326659.
Cf. A000522, A007526, A001339, A128195.

a:=[0];; for n in [2..25] do a[n]:=(n-1)*(a[n-1]+1); od; a; # Muniru A Asiru, Aug 07 2018

a007526 n = a007526_list !! n
a007526_list = 0 : zipWith (*) [1..] (map (+ 1) a007526_list)
-- Reinhard Zumkeller, Aug 27 2013

A007526 := n -> add(n!/k!,k=0..n) - 1;
a := n -> n*hypergeom([1,1-n],[],-1):
seq(simplify(a(n)), n=0..22); # Peter Luschny, May 09 2017
# third Maple program:
a:= proc(n) option remember;
      `if`(n<0, 0, n*(1+a(n-1)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jan 06 2020

Table[ Sum[n!/(n - r)!, {r, 1, n}], {n, 0, 20}] (* or *) Table[n!*Sum[1/k!, {k, 0, n - 1}], {n, 0, 20}]
a=1;Table[a=(a-1)*(n-1);Abs[a],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 20 2009 *)
FoldList[#1*#2 + #2 &, 0, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
f[n_] := Floor[E*n! - 1]; f[0] = 0; Array[f, 20, 0] (* Robert G. Wilson v, Feb 06 2015 *)
a[n_] := n (a[n - 1] +1); a[0] = 0; Array[a, 20, 0] (* Robert G. Wilson v, Feb 06 2015 *)
Round@Table[E n Gamma[n, 1], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 07 2016 *)

{a(n) = if( n<1, 0, n * (a(n-1) + 1))}; /* Michael Somos, Apr 06 2003 */

{a(n) = if( n<0, 0, n! * polcoeff(x * exp(x + x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Mar 04 2004 */

PARI
```
a(n)= sum(k=1,n, prod(j=0,k-1,n-j))
```

a(n) = A000522(n) - 1.
a(n) = floor(e*n! - 1). - Joseph K. Horn
a(n) = Sum_{r=1..n} A008279(n, r)= n!*(Sum_{k=0..n-1} 1/k!).
a(n) = n*(a(n-1) + 1).
E.g.f.: x*exp(x)/(1-x). - Vladeta Jovovic, Aug 25 2002
a(n) = Sum_{k=1..n} k!*C(n, k). - Benoit Cloitre, Dec 06 2002
a(n) = Sum_{k=0..n-1} (n! / k!). - Ross La Haye, Sep 22 2004
a(n) = Sum_{k=1..n} (Product_{j=0..k-1} (n-j)). - Joerg Arndt, Apr 24 2011
Binomial transform of n! - !n. - Paul Barry, May 12 2004
Inverse binomial transform of A066534. - Ross La Haye, Sep 16 2004
For n > 0, a(n) = exp(1) * Integral_{x>=0} exp(-exp(x/n)+x) dx. - Gerald McGarvey, Oct 19 2006
a(n) = Integral_{x>=0} (((1+x)^n-1)*exp(-x)). - Paul Barry, Feb 06 2008
a(n) = GAMMA(n+2)*(1+(-GAMMA(n+1)+exp(1)*GAMMA(n+1, 1))/GAMMA(n+1)). - Thomas Wieder, May 02 2009
E.g.f.: -1/G(0) where G(k) = 1 - 1/(x - x^3/(x^2+(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 10 2012
Conjecture : a(n) = (n+2)*a(n-1) - (2*n-1)*a(n-2) + (n-2)*a(n-3). - R. J. Mathar, Dec 04 2012 [Conjecture verified by Robert FERREOL, Aug 04 2018]
G.f.: (Q(0) - 1)/(1-x), where Q(k)= 1 + (2*k + 1)*x/( 1 - x - 2*x*(1-x)*(k+1)/(2*x*(k+1) + (1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 09 2013
G.f.: 2/((1-x)*G(0)) - 1/(1-x), where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+3) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
a(n) = (...((((((0)+1)*1+1)*2+1)*3+1)*4+1)...*n). - Bob Selcoe, Jul 04 2013
G.f.: Q(0)/(2-2*x) - 1/(1-x), where Q(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013
G.f.: (W(0) - 1)/(1-x), where W(k) = 1 - x*(k+1)/( x*(k+2) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013
For n > 0: a(n) = n*A000522(n-1). - Reinhard Zumkeller, Aug 27 2013
a(n) = (...(((((0)*1+1)*2+2)*3+3)*4+4)...*n+n). - Bob Selcoe, Apr 30 2014
0 = 1 + a(n)*(+1 + a(n+1) - a(n+2)) + a(n+1)*(+2 +a(n+1)) - a(n+2) for all n >= 0. - Michael Somos, Aug 30 2016
a(n) = n*hypergeom([1, 1-n], [], -1). - Peter Luschny, May 09 2017
Product_{n>=1} (a(n)+1)/a(n) = e, coming from Product_{n=1..N}(a(n)+1)/a(n) = Sum_{n=0..N} 1/n!. - Robert FERREOL, Jul 12 2018
O.g.f.: Sum_{k>=1} k^k*x^k/(1 + (k - 1)*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018

A128196 a(n) = (2n - 1)a(n - 1) + 2^n for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 3, 13, 73, 527, 4775, 52589, 683785, 10257031, 174370039, 3313031765, 69573669113, 1600194393695, 40004859850567, 1080131215981693, 31323805263501865, 971037963168623351, 32044252784564701655, 1121548847459764820069, 41497307356011298866841, 1618394986884440656855375

Offset: 0

Peter Luschny, Feb 26 2007

A weighted sum of quotients of double factorials.

a(n) are the row sum of triangle A126063.

P. Luschny, Variants of Variations.

Cf. A128195, A001147, A126063.

a := n -> `if`(n=0,1,(2*n-1)*a(n-1)+2^n);

a[n_] := Sum[2^k*((2*n-1)!!/(2*k-1)!!), {k, 0, n}]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 28 2013 *)

a(n) = (2n)!/(n! 2^n) Sum(k=0..n, 4^k k!/(2k)!)
a(n) = 2^n Gamma(n+1/2) Sum(k=0..n, 1/Gamma(k+1/2))
a(n) = Sum(k=0..n, 2^k n!!/k!!) [n!! defined as A001147(n), Gottfried Helms]
a(n) = Sum(k=0..n, 2^(2k-n)((n+1)! Catalan(n))/((k+1)! Catalan(k))) [Catalan(n) A000108]
a(n) = Sum(k=0..n, 2^(2k-n) QuadFact(n)/QuadFact(k)) [QuadFact(n) A001813]
a(n) = Sum(k=0..n, 2^(2k-n) (-1)^(n-k) A097388(n)/A097388(k) )
a(n) = A001147(n) Sum(k=0..n, 2^k / A001147(k))
a(n) = A128195(n)/A005408(n)
a(n) = A128195(n-1)+A000079(n) (if n>0)
Recursive form: a(n) = (2n-1)*a(n-1) + 2^n; a(0) = 1 [Gottfried Helms]
Note: The following constants will be used in the next formulas.
K = (1-exp(1)*Gamma(1/2,1))/Gamma(1/2)
M = sqrt(2)(1+exp(1)(Gamma(1/2)-Gamma(1/2,1)))
Generalized form: For x>0
a(x) = 2^x(exp(1)*Gamma(x+1/2,1) + K*Gamma(x+1/2))
Asymptotic formula:
a(n) ~ 2^n*(1+(exp(1)+K)*(n-1/2)!)
a(n) ~ M(2exp(-1)(n-1/(24*n+19/10*1/n)))^n

cp's OEIS Frontend

A126062 Array read by antidiagonals: see A128195 for details.

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A005917 Rhombic dodecahedral numbers: a(n) = n^4 - (n - 1)^4.

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A007526 a(n) = n*(a(n-1) + 1), a(0) = 0.

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A128196 a(n) = (2*n - 1)*a(n - 1) + 2^n for n >= 1, a(0) = 1.

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A128198 Array read by antidiagonals. A scheme of arrangements: ArrScheme(k,n) = VarScheme(k,n-1) + k^n; ArrScheme(k,0) = 1. VarScheme(k,n) = (n*k+1)*(VarScheme(k,n-1) + k^n); VarScheme(k,0) = 1.

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A128196 a(n) = (2n - 1)a(n - 1) + 2^n for n >= 1, a(0) = 1.

A128198 Array read by antidiagonals. A scheme of arrangements: ArrScheme(k,n) = VarScheme(k,n-1) + k^n; ArrScheme(k,0) = 1. VarScheme(k,n) = (nk+1)(VarScheme(k,n-1) + k^n); VarScheme(k,0) = 1.