A001792 -id:A001792 - cp's OEIS frontend

1, 3, 1, 8, 4, 1, 20, 12, 5, 1, 48, 32, 17, 6, 1, 112, 80, 49, 23, 7, 1, 256, 192, 129, 72, 30, 8, 1, 576, 448, 321, 201, 102, 38, 9, 1, 1280, 1024, 769, 522, 303, 140, 47, 10, 1, 2816, 2304, 1793, 1291, 825, 443, 187, 57, 11, 1, 6144, 5120, 4097, 3084, 2116, 1268, 630

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((1-z)/(1-2*z)^2)/(1-x*z/(1-z)).
This is the second member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
The column sequences appear in A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 for m=0..7.

			1;
3,1;
8,4,1;
20,12,5,1;
...
Fourth row polynomial (n=3): p(3,x)= 20+12*x+5*x^2+x^3

G. C. Greubel, Table of n, a(n) for n = 0..1274

Cf. A007318, A055248, A008949. Row sums: A049611(n+1) = A055252(n, 0).

a[n_, m_] := Binomial[n, m]*Hypergeometric2F1[2, m-n, m+1, -1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 11 2014 *)

a(n, m) = Sum_{k=m,..,n} ( A055248(n, k) ), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m) = Sum_{j=m,..,(n-1)} ( a(j, m) ) + A055248(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: ((1-x)/(1-2*x)^2)*(x/(1-x))^m, m >= 0.

a(n, m) = binomial(n, m) * 2F1(2, m-n; m+1; -1) where 2F1 is the hypergeometric function. Jean-François Alcover, Mar 11 2014

A066810 Expansion of x^2/((1-3x)(1-2*x)^2).

Original entry on oeis.org

0, 0, 1, 7, 33, 131, 473, 1611, 5281, 16867, 52905, 163835, 502769, 1532883, 4651897, 14070379, 42456897, 127894979, 384799049, 1156756443, 3475250065, 10436235955, 31330727961, 94038321227, 282211432673, 846835624611, 2540926304233, 7623651327931, 22872765923121

Offset: 0

N. J. A. Sloane, Jan 25 2002

Binomial transform of A000295.
a(n) = A112626(n, 2). - Ross La Haye, Jan 11 2006
Let Q be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all x,y of P(A), xQy if x is a proper subset of y and |y| - |x| > 1. Then a(n) = |Q|. - Ross La Haye, Jan 11 2008
a(n) is the number of n-digit ternary sequences that have at least two 0's. - Geoffrey Critzer, Apr 14 2009

Harry J. Smith, Table of n, a(n) for n = 0..200
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (7,-16,12).

Column k=1 of A238858 (with different offset).

List([0..30], n-> 3^n - 2^n - n*2^(n-1)); # G. C. Greubel, Nov 18 2019

[3^n-2^n-n*2^(n-1): n in [0..30]]; // Vincenzo Librandi, Nov 29 2015

seq(3^n - 2^n - n*2^(n-1), n=0..30); # G. C. Greubel, Nov 18 2019

RecurrenceTable[{a[n]==3*a[n-1] + (n-1) 2^(n-2), a[0]==0}, a, {n, 0, 30}] (* Geoffrey Critzer, Apr 14 2009 *)
CoefficientList[Series[x^2/((1-3x)(1-2x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 29 2015 *)

a(n) = 3^n -2^n -n*2^(n-1) \\ Harry J. Smith, Mar 29 2010

[3^n - 2^n - n*2^(n-1) for n in (0..30)] # G. C. Greubel, Nov 18 2019

a(n) = 3^n - 2^n - n*2^(n-1).
From Ross La Haye, Apr 26 2006: (Start)
a(n) = A000244(n) - A001792(n).
a(n) = Sum_{k=2..n} binomial(n,k)2^(n-k). (End)
Inverse binomial transform of A086443. - Ross La Haye, Apr 29 2006
Convolution of A000244 beginning [0,1,3,9,27,81,...] and A001787. - Ross La Haye, Feb 15 2007
From Geoffrey Critzer, Apr 14 2009: (Start)
E.g.f.: exp(2*x)*(exp(x) - x - 1).
a(n) = 3*a(n-1) + (n-1)*2^(n-2). (End)

Additional comments from Ross La Haye, Sep 27 2005

A162326 Let a(0) = a(1) = 1, and na(n) = 2(-7+5n)a(n-1) + 9(2-n)a(n-2) for n >= 2.

Original entry on oeis.org

1, 1, 3, 13, 71, 441, 2955, 20805, 151695, 1135345, 8671763, 67320573, 529626839, 4213228969, 33833367963, 273892683573, 2232832964895, 18314495896545, 151037687326755, 1251606057754605, 10416531069771111, 87029307323766681

Offset: 0

Georg Muntingh, Jul 01 2009

nonn

Let y = y(x) be implicitly defined by g(x,y(x)) = 0, with dg/dy not identically zero. For n >= 1, the sequence a(n) is the number of terms in the expansion of the divided difference [x0,...,xn]y in terms of bivariate divided differences of g.
(1 + 3*x + 13*x^2 + 71*x^3 + ...) = (1 + 4*x + 20*x^2 + 116*x^3 + ...) * 1/(1 + x + 4*x^2 + 20*x^3 + 116*x^4 + ...); where A082298 = (1, 4, 20, 116, 740, ...). - Gary W. Adamson, Nov 17 2011
The shifted sequence 1,3,13,71,... is the binomial transform of A151374. - Georg Muntingh, Jul 19 2012
a(n+1) is the number of Schröder paths of semilength n in which the (2,0)-steps come in 3 colors and with no peaks at level 1. - José Luis Ramírez Ramírez, Mar 31 2013
Define an infinite triangle by T(n,0)=1 and the other cells by T(n,k) = Sum_{c=0..k-1} T(n,c) + Sum_{r=k..n-1} T(r,k), the sum of the cells to the left and above a cell. The column k=1 contains A000079, the column k=2 essentially A001792. Then T(n,n)=a(n) on the diagonal. - J. M. Bergot, May 22 2013

			Write [0...n]y for [x0,...,xn]y and [0...s,0...t]g for [x0,...,xs;y0,...,yt]g.
For n = 1 one finds 1 term, [01]y = -[01;1]g/[0;01]g.
For n = 2 one finds 3 terms, [012]y = -[012;2]g/[0;02]g + ([01;12]g[12;2]g)/([0;02]g[1;12]g) - ([0;012]g[01;1]g[12;2]g)/([0;02]g[0;01]g[1;12]g).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Georg Muntingh, Implicit Divided Differences, Little Schroeder Numbers, and Catalan Numbers, J. Integ. Seqs., Vol. 15 (2012), Article 12.6.5.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 98.

Cf. A172003, which is a generalization to bivariate implicit functions.
Cf. A003262, which is the analogous sequence for implicit derivatives, and A172004 for its generalization to bivariate implicit functions.
Cf. A082298, A000108, A151374.

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (5-Sqrt((1-9*x)/(1-x)))/4 )); // G. C. Greubel, Feb 07 2019

a:=[1,3]; for n in [3..21] do Append(~a,(2*(-7+5*n)*a[n-1] + 9*(2-n)*a[n-2]) div n); end for ; [1] cat a; // Marius A. Burtea, Jan 20 2020

CoefficientList[Series[(5-Sqrt[(1-9*x)/(1-x)])/4, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n):=if n=0 then 1 else sum(binomial(n,k)*2^(n-k-1)*binomial(2*n-2*k-2,n-k-1),k,0,n)/n; /* Vladimir Kruchinin, Mar 13 2016 */

a(n) = if(n<2, 1, (2*(-7+5*n)*a(n-1) + 9*(2-n)*a(n-2))/n);
vector(25, n, a(n-1)) \\ Altug Alkan, Oct 06 2015

my(x='x+O('x^20)); Vec((5-sqrt((1-9*x)/(1-x)))/4) \\ G. C. Greubel, Feb 07 2019

L = [1, 1]
for n in range(2,22):
    L.append( ((-14 + 10*n)*L[-1] + (18-9*n)*L[-2])//n )
print(L)
# Georg Muntingh, Jul 19 2012

((5-sqrt((1-9*x)/(1-x)))/4).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Feb 07 2019

Let E = N x N \ {(0,0), (0,1)} be a set of pairs of natural numbers. The number of terms a(n) is the coefficient of x^n*y^{n-1} of the generating function 1 - log(1 - Sum_{(s,t) in E} x^s*y^{s+t-1}) = 1 + Sum_{q >= 1} (Sum_{(s,t) in E} x^s*y^{s+t-1})^q / q.
From Georg Muntingh, Jul 19 2012: (Start)
a(n) = 2F1(1/2,1-n;2;-8), where 2F1 is the Gauss hypergeometric series.
G.f.: (5 - sqrt( (1-9*x)/(1-x) ))/4.
Quadratic recurrence relation: a(n) = 1 + 2*Sum_{m=1..n-1} a(m)*a(n-m).
(End)
a(n) ~ 3^(2*n+1)/(16*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
a(n) = Sum_{k=0..n} (binomial(n,k)*2^(n-k-1)*binomial(2*n-2*k-2,n-k-1))/n, n>0, a(0)=1. - Vladimir Kruchinin, Mar 13 2016
From Peter Bala, Jan 19 2020: (Start)
a(n+1) = Sum_{k = 0..n} 2^k*C(n,k)*Catalan(k).
a(n+1) = (2/Pi) * Integral_{x = -1..1} (1 + 8*x^2)^n*sqrt(1 - x^2) dx.
O.g.f.: 1 + x/(1 - x)*c(2*x/(1-x)), where c(x) is the o.g.f. for A000108. (End)
Conjecture: a(n) = t_n for n > 0 with a(0) = 1 where we start with vector v of fixed length m with elements v_i = 1, then set t = v and for i=1..m-1, for j=i+1..m apply [v_i, v_j] := [v_i + 2*v_j, 2*v_i + v_j] (here square brackets mean that instead of sequentially assigning v_i and then v_j, we reserve their values (for example, as A = v_i, B = v_j) and then assign them in any order) and t_{i+1} := v_{i+1} (after ending each cycle for j). It also looks like that if we change 2*v_i to z*v_i it gives us a(n+1) = Sum_{k=0..n} A090981(n, k)*2^(n-k) for n >= 0. - Mikhail Kurkov, Aug 14 2024

Edited by Georg Muntingh, Jan 22 2010

A039717 Row sums of convolution triangle A030523.

Original entry on oeis.org

1, 4, 15, 55, 200, 725, 2625, 9500, 34375, 124375, 450000, 1628125, 5890625, 21312500, 77109375, 278984375, 1009375000, 3651953125, 13212890625, 47804687500, 172958984375, 625771484375, 2264062500000, 8191455078125

Offset: 1

Wolfdieter Lang

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 5.
With offset 0 = INVERT transform of A001792: (1, 3, 8, 20, 48, 112, ...). - Gary W. Adamson, Oct 26 2010
From Tom Copeland, Nov 09 2014: (Start)
The array belongs to a family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the o.g.f. (1-sqrt(1-4x/(1+(1-t)x)))/2 and inverse x*(1-x)/(1 + (t-1)*x*(1-x)). See A091867 for more info on this family. Here t = -4 (mod signs in the results).
Let C(x) = (1 - sqrt(1-4x))/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1-x) and P(x,t) = x/(1+t*x) with inverse P(x,-t).
O.g.f.: G(x) = x*(1-x)/(1 - 5x*(1-x)) = P(Cinv(x),-5).
Inverse O.g.f.: Ginv(x) = (1 - sqrt(1 - 4*x/(1+5x)))/2 = C(P(x,5)) (signed A026378). Cf. A030528. (End)
p-INVERT of (2^n), where p(s) = 1 - s - s^2; see A289780. - Clark Kimberling, Aug 10 2017

Michel Marcus, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Appears in A109106.

Cf. A000045, A000108, A001792, A005043, A091867, A026378, A030528.

CoefficientList[Series[(1 - x) / (1 - 5 x + 5 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)

Vec(x*(1-x)/(1-5*x+5*x^2) + O(x^40)) \\ Altug Alkan, Nov 20 2015

G.f.: x*(1-x)/(1-5*x+5*x^2) = g1(3, x)/(1-g1(3, x)), g1(3, x) := x*(1-x)/(1-2*x)^2 (g.f. first column of A030523).
From Paul Barry, Apr 16 2004: (Start)
Binomial transform of Fibonacci(2n+2).
a(n) = (sqrt(5)/2 + 5/2)^n*(3*sqrt(5)/10 + 1/2) - (5/2 - sqrt(5)/2)^n*(3*sqrt(5)/10 - 1/2). (End)
a(n) = (1/5)*Sum_{r=1..9} sin(3*r*Pi/10)*sin(r*Pi/2)*(2*cos(r*Pi/10))^(2*n).
a(n) = 5*a(n-1) - 5*a(n-2).
a(n) = Sum_{k=0..n} Sum_{i=0..n} binomial(n, i)*binomial(k+i+1, 2k+1). - Paul Barry, Jun 22 2004
From Johannes W. Meijer, Jul 01 2010: (Start)
Limit_{k->oo} a(n+k)/a(k) = (A020876(n) + A093131(n)*sqrt(5))/2.
Limit_{n->oo} A020876(n)/A093131(n) = sqrt(5).
(End)
From Benito van der Zander, Nov 19 2015: (Start)
Limit_{k->oo} a(k+1)/a(k) = 1 + phi^2 = (5 + sqrt(5)) / 2.
a(n) = a(n-1) * 3 + A081567(n-2) (not proved).
(End)
E.g.f.: exp(x*5/2) * (cosh(x*sqrt(5)/2) + (3/sqrt(5))*sinh(x*sqrt(5)/2)). - Fabian Pereyra, Oct 29 2024

A087447 a(0) = a(1) = 1; for n > 1, a(n) = (n+2)*2^(n-2).

Original entry on oeis.org

1, 1, 4, 10, 24, 56, 128, 288, 640, 1408, 3072, 6656, 14336, 30720, 65536, 139264, 294912, 622592, 1310720, 2752512, 5767168, 12058624, 25165824, 52428800, 109051904, 226492416, 469762048, 973078528, 2013265920, 4160749568

Offset: 0

Paul Barry, Sep 05 2003

Binomial transform of A005408 (with interpolated zeros). Binomial transform is A087448. a(n+2) = 2*A045623(n+1); a(n+1) = A001792(n) + (0^n - (-2)^n)/2. The sequence 1,4,10,... given by 2^n*(n+3)/2 - 0^n/2 is the binomial transform of 1,3,3,5,5,...
Equals real part of binomial transform of [1, 2*i, 3, 4*i, 5, 6*i, ...]; i=sqrt(-1). - Gary W. Adamson, Sep 21 2008
An elephant sequence, see A175655. For the central square 24 A[5] vectors, with decimal values between 27 and 432, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A057711 (without the leading 0). - Johannes W. Meijer, Aug 15 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Essentially same as A079859.

Cf. A001792, A005408, A045623, A057711, A087448, A175655.

Join[{1, 1}, Table[(n + 2) 2^(n - 2), {n, 2, 30}]]  (* Harvey P. Dale, Feb 22 2011 *)

def A087447(n): return n+2<1 else 1 # Chai Wah Wu, Oct 03 2024

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*(2k+1). - Paul Barry, Nov 29 2004
From Colin Barker, Mar 23 2012: (Start)
G.f.: (1-x)*(1-2*x+2*x^2)/(1-2*x)^2.
a(n) = 4*a(n-1) - 4*a(n-2) for n > 3. (End)
E.g.f.: (1 - x + exp(2*x)*(1 + x))/2. - Stefano Spezia, Jan 31 2023

Definition corrected (by a factor of 2) by R. J. Mathar, Feb 21 2009

A003583 a(n) = (n+2)2^(2n-1) - (n/2)binomial(2n,n).

Original entry on oeis.org

1, 5, 26, 130, 628, 2954, 13612, 61716, 276200, 1223002, 5367676, 23383100, 101215576, 435712580, 1866667448, 7963424104, 33846062544, 143373104378, 605518549660, 2550438016812, 10716162617336

Offset: 0

N. J. A. Sloane

nonn

a(n) gives the number of open partitions of a tree made of two chains with n points each, that share an added root. (An open partition pi of a tree T is a partition of the vertices of T with the property that, for each block B of pi, the upset of B is a union of blocks of pi.) - Pietro Codara, Jan 14 2015

Pietro Codara, Partitions of a finite partially ordered set, From Combinatorics to Philosophy: The Legacy of G.-C. Rota, Springer, New York (2009), 45-59.

G. C. Greubel, Table of n, a(n) for n = 0..1000
N. J. Calkin, A curious binomial identity, Discr. Math., 131 (1994), 335-337.
Pietro Codara, Ottavio D'Antona, Francesco Marigo, Corrado Monti, Making simple proofs simpler, arXiv:1307.1348 [cs.MS], 2013.
Zachary Hamaker, Eric Marberg, Atoms for signed permutations, arXiv:1802.09805 [math.CO], 2018.
M. Hirschhorn, Calkin's binomial identity, Discr. Math., 159 (1996), 273-278.
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

If the exponent E in a(n) = Sum_{m=0..n} (Sum_{k=0..m} C(n,k))^E is 1, 2, 3, 4, 5 we get A001792, A003583, A007403, A294435, A294436 respectively.

seq((n+2)*2^(2*n-1)-(n/2)*binomial(2*n,n), n=0..50); # Robert Israel, Jan 13 2015

Table[(n+2)*2^(2*n-1)-(n/2)*Binomial[2*n,n], {n,0,50}] (* Pietro Codara, Jan 14 2015 *)
Table[Sum[Sum[Binomial[n-1,k-1]Binomial[n-1,j-1]Min[k,j],{j,1,n}],{k,1 n}],{n,1,51}] (* Pietro Codara, Jan 14 2015 *)

x='x+O('x^50); Vec((1-2*x)/(1-4*x)^2 - x/(1 - 4*x)^(3/2)) \\ G. C. Greubel, Feb 15 2017

Main diagonal of correlation matrix of A055248. a(n) = Sum_{k=0..n} ( Sum_{m=k..n} binomial(n, m) )^2. - Paul Barry, Jun 05 2003
Let S2 := (n, t)->add( k^t * (add( binomial(n, j), j=0..k))^2, k=0..n); a(n) = S2(n, 0).
From Robert Israel, Jan 13 2015: (Start)
G.f.: (1-2*x)/(1-4*x)^2 - x/(1 - 4*x)^(3/2).
E.g.f.: (2*x+1)*exp(4*x) - x*exp(2*x)*(I_0(2*x)+I_1(2*x)) where I_0 and I_1 are modified Bessel functions.
a(n) ~ 4^n*(n/2 - sqrt(n)/(2*sqrt(Pi)) + 1 + O(n^(-1/2))).
(End)

A033842 Triangle of coefficients of certain polynomials (exponents in decreasing order).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 16, 16, 6, 1, 125, 125, 50, 10, 1, 1296, 1296, 540, 120, 15, 1, 16807, 16807, 7203, 1715, 245, 21, 1, 262144, 262144, 114688, 28672, 4480, 448, 28, 1, 4782969, 4782969, 2125764, 551124, 91854, 10206, 756, 36, 1, 100000000

Offset: 0

Wolfdieter Lang

See A049323.

			{1}; {1,1}; {3,3,1}; {16,16,6,1}; {125,125,50,10,1}; .... E.g. third row {3,3,1} corresponds to polynomial p(2,x)= 3*x^2+3*x+1.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Thierry Lévy, The Number of Prefixes of Minimal Factorisations of a Cycle, The Electronic Journal of Combinatorics, 23(3) (2016), #P3.35

a(n, 0)= A000272(n+1), n >= 0 (first column), a(n, 1)= A000272(n+1), n >= 1 (second column). p(k-1, -x)/(1-k*x)^k = (-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.

A035341 Sum of ordered factorizations over all prime signatures with n factors.

Original entry on oeis.org

1, 1, 5, 25, 173, 1297, 12225, 124997, 1492765, 19452389, 284145077, 4500039733, 78159312233, 1460072616929, 29459406350773, 634783708448137, 14613962109584749, 356957383060502945, 9241222160142506097, 252390723655315856437, 7260629936987794508973

Offset: 0

Alford Arnold

Let f(n) = number of ordered factorizations of n (A074206(n)); a(n) = sum of f(k) over all terms k in A025487 that have n factors.
When the unordered spectrum A035310 is so ordered the sequences A000041 A000070 ...A035098 A000110 yield A000079 A001792 ... A005649 A000670 respectively.
Row sums of A095705. - David Wasserman, Feb 22 2008
From Ludovic Schwob, Sep 23 2023: (Start)
a(n) is the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums. The a(3) = 25 matrices:
  [1 1 1] [1 2] [2 1] [3]
  .
  [1 1] [1 1] [1 1 0] [1 0 1] [0 1 1] [2] [0 2] [2 0]
  [1 0] [0 1] [0 0 1] [0 1 0] [1 0 0] [1] [1 0] [0 1]
  .
  [1] [1 0] [0 1] [1 0] [0 1] [1 0 0] [1 0 0] [0 1] [1 0]
  [1] [1 0] [0 1] [0 1] [1 0] [0 1 0] [0 0 1] [1 0] [0 1]
  [1] [0 1] [1 0] [1 0] [0 1] [0 0 1] [0 1 0] [1 0] [0 1]
  .
  [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [1 0 0] [0 1 0] [1 0 0] (End)

			a(3) = 25 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=4, f(12)=8, f(30)=13 and 4+8+13 = 25.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 36 terms from David Wasserman)
Eric Weisstein's World of Mathematics, Perfect Partition

Cf. A005651, A025487, A035310, A255906, A365961.

Row sums of A261719.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*If[j == 0, 1, Binomial[i + k - 1, k - 1]^j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz, updated Dec 15 2020 *)

R(n,k)=Vec(-1 + 1/prod(j=1, n, 1 - binomial(k+j-1,j)*x^j + O(x*x^n)))
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023

a(n) ~ c * n! / log(2)^n, where c = 1/(2*log(2)) * Product_{k>=2} 1/(1-1/k!) = A247551 / (2*log(2)) = 1.8246323... . - Vaclav Kotesovec, Jan 21 2017

More terms from Erich Friedman.

More terms from David Wasserman, Feb 22 2008

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cp's OEIS Frontend

A083711 a(n) = A083710(n) - A000041(n-1).

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A089658 a(n) = S1(n,1), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

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A055249 Triangle of partial row sums (prs) of triangle A055248 (prs of Pascal's triangle A007318).

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A066810 Expansion of x^2/((1-3*x)*(1-2*x)^2).

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A162326 Let a(0) = a(1) = 1, and n*a(n) = 2*(-7+5*n)*a(n-1) + 9*(2-n)*a(n-2) for n >= 2.

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A039717 Row sums of convolution triangle A030523.

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A066810 Expansion of x^2/((1-3x)(1-2*x)^2).

A162326 Let a(0) = a(1) = 1, and na(n) = 2(-7+5n)a(n-1) + 9(2-n)a(n-2) for n >= 2.

A003583 a(n) = (n+2)2^(2n-1) - (n/2)binomial(2n,n).