A035009 -id:A035009 - cp's OEIS frontend

A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

1, 2, 3, 5, 4, 7, 10, 15, 5, 9, 13, 20, 17, 27, 37, 52, 6, 11, 16, 25, 21, 34, 47, 67, 26, 43, 60, 87, 77, 114, 151, 203, 7, 13, 19, 30, 25, 41, 57, 82, 31, 52, 73, 107, 94, 141, 188, 255, 37, 63, 89, 132, 115, 175, 235, 322, 141, 218, 295, 409, 372, 523, 674

Offset: 0

Mikhail Kurkov, Aug 23 2021 [verification needed]

Modulo 2 binomial transform of A243499(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Kevin Ryde, PARI/GP Code

Cf. A000110, A000120, A002720, A005493, A005494, A007814, A008277, A035009, A045379, A053645, A129760, A143494, A143495, A196834, A265649, A295989.

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

function a = A347204(max_n)
    a(1) = 1;
    a(2) = 2;
    for nloop = 3:max_n
        n = nloop-1;
        s = 0;
        for k = 0:floor(log2(n))-1
            s = s + a(1+A053645(n)-2^k*(mod(floor(n/(2^k)),2)));
        end
        a(nloop) = 2*a(A053645(n)+1) + s;
    end
end
function a_n = A053645(n)
    a_n = n - 2^floor(log2(n));
end % Thomas Scheuerle, Oct 25 2021

f[n_] := BitAnd[n, n - 1]; a[0] = 1; a[n_] := a[n] = a[f[n]/2] + a[Floor[(n + f[n])/2]]; Array[a, 100, 0] (* Amiram Eldar, Nov 19 2021 *)

f(n) = bitand(n, n-1); \\ A129760
a(n) = if (n<=1, n+1, if (n%2, a(n\2)+a(n-1), a(f(n/2)) + a(n/2+f(n/2)))); \\ Michel Marcus, Oct 25 2021

PARI
```
\\ Also see links.
```

A129760(n) = bitand(n, n-1);
memoA347204 = Map();
A347204(n) = if (n<=1, n+1, my(v); if(mapisdefined(memoA347204,n,&v), v, v = if(n%2, A347204(n\2)+A347204(n-1), A347204(A129760(n/2)) + A347204(n/2+A129760(n/2))); mapput(memoA347204,n,v); (v))); \\ (Memoized version of Michel Marcus's program given above) - Antti Karttunen, Nov 20 2021

a(n) = a(n - 2^f(n)) + (1 + f(n))*a((n - 2^f(n))/2) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(2n+1) = a(n) + a(2n) for n >= 0.
a(2n) = a(n - 2^f(n)) + a(2n - 2^f(n)) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(n) = 2*a(f(n)) + Sum_{k=0..floor(log_2(n))-1} a(f(n) - 2^k*T(n,k)) for n > 1 with a(0) = 1, a(1) = 2, and where f(n) = A053645(n), T(n,k) = floor(n/2^k) mod 2.
Sum_{k=0..2^n - 1} a(k) = A035009(n+1) for n >= 0.
a((4^n - 1)/3) = A002720(n) for n >= 0.
a(2^n - 1) = A000110(n+1),
a(2*(2^n - 1)) = A005493(n),
a(2^2*(2^n - 1)) = A005494(n),
a(2^3*(2^n - 1)) = A045379(n),
a(2^4*(2^n - 1)) = A196834(n),
a(2^m*(2^n-1)) = T(n,m+1) is the n-th (m+1)-Bell number for n >= 0, m >= 0 where T(n,m) = m*T(n-1,m) + Sum_{k=0..n-1} binomial(n-1,k)*T(k,m) with T(0,m) = 1.
a(n) = Sum_{j=0..2^A000120(n)-1} A243499(A295989(n,j)) for n >= 0. Also A243499(n) = Sum_{j=0..2^f(n)-1} (-1)^(f(n)-f(j)) a(A295989(n,j)) for n >= 0 where f(n) = A000120(n). In other words, a(n) = Sum_{j=0..n} (binomial(n,j) mod 2)*A243499(j) and A243499(n) = Sum_{j=0..n} (-1)^(f(n)-f(j))*(binomial(n,j) mod 2)*a(j) for n >= 0 where f(n) = A000120(n).
Generalization:
b(n, x) = (1/x)*b((n - 2^f(n))/2, x) + (-1)^n*b(floor((2n - 2^f(n))/2), x) for n > 0 with b(0, x) = 1 where f(n) = A007814(n).
Sum_{k=0..2^n - 1} b(k, x) = (1/x)^n for n >= 0.
b((4^n - 1)/3, x) = (1/x)^n*n!*L_{n}(x) for n >= 0 where L_{n}(x) is the n-th Laguerre polynomial.
b((8^n - 1)/7, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A265649(n, k) for n >= 0.
b(2^n - 1, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A008277(n+1, k+1),
b(2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143494(n+2, k+2),
b(2^2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143495(n+3, k+3),
b(2^m*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*T(n+m+1, k+m+1, m+1) for n >= 0, m >= 0 where T(n,k,m) is m-Stirling numbers of the second kind.

A355247 Expansion of e.g.f. exp(2*(exp(x) - 1 + x)).

Original entry on oeis.org

1, 4, 18, 90, 494, 2946, 18926, 130066, 950654, 7353794, 59954638, 513333618, 4601380766, 43062556322, 419742815726, 4252083713874, 44680229906622, 486145710591874, 5468499473222670, 63503107472489266, 760281866742088670, 9373065303624742498, 118858898763010225198

Offset: 0

Vaclav Kotesovec, Jun 25 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..550

Cf. A000110, A001861, A035009, A194689, A217924, A293024, A339014.

nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+2*x], {x, 0, nmax}], x] * Range[0, nmax]!
Table[(BellB[n+2, 2] - BellB[n+1, 2])/4, {n, 0, 25}] (* Vaclav Kotesovec, Jul 21 2025 *)

a(n) ~ n^(n+2) * exp(n/LambertW(n/2) - n - 2) / (4 * sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n+2)).

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k+1) * Bell(n-k+1). - Ilya Gutkovskiy, Jun 26 2022

A335980 Expansion of e.g.f. exp(2 * (1 - exp(-x)) + x).

Original entry on oeis.org

1, 3, 7, 11, 7, -5, 23, 75, -281, -101, 4663, -14229, -41721, 532667, -1464489, -8840053, 103689511, -313202725, -2348557705, 32041266859, -127039882425, -762423051013, 14393151011735, -81523161874741, -236027974047897, 8564406463119387

Offset: 0

Ilya Gutkovskiy, Jul 03 2020

sign

Cf. A001861, A035009, A109747, A213170, A335868, A335981, A335982.

nmax = 25; CoefficientList[Series[Exp[2 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 2 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]

my(N=33, x='x+O('x^N)); Vec(serlaplace(exp(2 * (1 - exp(-x)) + x))) \\ Joerg Arndt, Jul 04 2020

a(n) = exp(2) * (-1)^n * Sum_{k>=0} (-2)^k * (k - 1)^n / k!.

a(0) = 1; a(n) = a(n-1) + 2 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * a(k).

A355252 Expansion of e.g.f. exp(2(exp(x) - 1) + 3x).

Original entry on oeis.org

1, 5, 27, 157, 979, 6517, 46107, 345261, 2726243, 22623525, 196712171, 1787356765, 16929897395, 166808851541, 1706299041211, 18088031239437, 198392625389315, 2248104026019461, 26283054263021963, 316637825898555069, 3926250785070282579, 50056384077880370101

Offset: 0

Vaclav Kotesovec, Jun 26 2022

nonn

Binomial transform of A355247.

Vaclav Kotesovec, Table of n, a(n) for n = 0..550

Cf. A001861, A035009, A355247, A355253.

nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+3*x], {x, 0, nmax}], x] * Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1) + 3*x))) \\ Michel Marcus, Dec 04 2023

a(n) ~ n^(n+3) * exp(n/LambertW(n/2) - n - 2) / (8 * sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n+3)).

a(0) = 1; a(n) = 3 * a(n-1) + 2 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 04 2023

A367889 Expansion of e.g.f. exp(3(exp(x) - 1) + 2x).

Original entry on oeis.org

1, 5, 28, 173, 1165, 8468, 65923, 546197, 4791214, 44301143, 430158397, 4372004546, 46381674085, 512328076385, 5879362011436, 69958289731457, 861605015493073, 10965899141265500, 144018319806024991, 1949190279770578145, 27153595018237222774

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000110, A005493, A035009, A078940, A355247, A367888.

nmax = 20; CoefficientList[Series[Exp[3 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 2^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) + 2*x))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 + x * ( 2 * A(x) + 3 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k+2)^n / k!.
a(0) = 1; a(n) = 2 * a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A027710(k).

A029706 Sum C(n,k)*b(k), k=1..n, where b(k) is given by A001861.

Original entry on oeis.org

2, 10, 46, 226, 1214, 7106, 44958, 305090, 2206398, 16913986, 136823262, 1163490498, 10366252030, 96491364674, 935976996126, 9440144423874, 98800604237118, 1071092025420866, 12008090971866206, 139014305916844738

Offset: 0

Andre Poenitz (andre.poenitz(AT)mathematik.tu-chemnitz.de)

nonn

Jacques Carlier and Corinne Lucet, A decomposition algorithm for network reliability evaluation. In First International Colloquium on Graphs and Optimization (GOI), 1992 (Grimentz). Discrete Appl. Math. 65 (1996), 141-156 (see page 153 and Fig 6).

Arises if one of the two kinds of boxes mentioned in A001861 may 'fail'.

a[n_] := Sum[Binomial[n, k]*BellB[k, 2], {k, 1, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Apr 26 2013 *)

x='x+O('x^66); Vec(serlaplace(1/2-exp(x)+1/2*exp(2*exp(x)-2))) \\ Joerg Arndt, Apr 21 2013

a(n) = sum(k=1, n, binomial(n, k)*sum(j=1, k, stirling(k, j, 2)*2^j)); \\ \\ Michel Marcus, Apr 26 2013

E.g.f.: 1/2 - exp(x) + 1/2*exp(2*exp(x)-2). a(n-2) = A035009(n)-1. - Ralf Stephan, Jan 26 2004

Added missing digit in the last term from Jean-François Alcover, Apr 26 2013

A129340 Triangular array read by rows: for n, k >= 1, a(n+1, 1) = 2*a(n, n); a(n+1, k+1) = a(n, k)+a(n+1, k).

Original entry on oeis.org

1, 2, 3, 6, 8, 11, 22, 28, 36, 47, 94, 116, 144, 180, 227, 454, 548, 664, 808, 988, 1215, 2430, 2884, 3432, 4096, 4904, 5892, 7107, 14214, 16644, 19528, 22960, 27056, 31960, 37852, 44959, 89918, 104132, 120776, 140304, 163264, 190320, 222280

Offset: 1

Paul Curtz, May 28 2007

Main diagonal is A035009. First column is A001861.

Cf. A001861, A035009.

a(n, n) = A035009(n). For k < n, a(n, k) = 2*sum_{i = 1..k} binomial(k-1, i-1)*A035009(n-i).

Edited and extended by David Wasserman, May 02 2008

A324133 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 12 and t = 12.

Original entry on oeis.org

1, 1, 2, 6, 24, 114, 608, 3554, 22480, 152546, 1103200, 8456994, 68411632, 581745250, 5183126016, 48245682338, 467988498064, 4720072211938, 49400302118560, 535546012710434, 6004045485933104, 69507152958422370, 829789019700511040, 10202854323325253538, 129061753086335478736

Offset: 0

N. J. A. Sloane, Feb 16 2019

nonn

Stirling transform of j-> ceiling(2^(j-2)). - Alois P. Heinz, Aug 25 2021

Alois P. Heinz, Table of n, a(n) for n = 0..558
Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Cf. A000142, A001861, A035009, A347011.

b:= proc(n, m) option remember; `if`(n=0,
      ceil(2^(m-2)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Aug 25 2021

b[n_, m_] := b[n, m] = If[n == 0,
     Ceiling[2^(m-2)], m*b[n-1, m] + b[n-1, m+1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 15 2022, after Alois P. Heinz *)

a(n) = -2^(n-1) + 2*Sum_{i = 0..n-1} binomial(n-1,i) * a(i) with a(0) = 1. [It follows from Kitaev's recurrence for C_n on p. 220 of his paper.] - Petros Hadjicostas, Oct 30 2019
From Alois P. Heinz, Aug 25 2021: (Start)
G.f.: Sum_{k>=0} ceiling(2^(k-2))*x^k / Product_{j=1..k} (1-j*x).
a(n) = Sum_{j=0..n} Stirling2(n,j)*ceiling(2^(j-2)). (End)

More terms from Petros Hadjicostas, Oct 30 2019

A366199 Expansion of e.g.f. exp(4(exp(x) - 1) + 2x).

Original entry on oeis.org

1, 6, 40, 292, 2308, 19580, 177044, 1696572, 17148916, 182114972, 2024979604, 23506175868, 284125820724, 3567957972316, 46454893734612, 625979771144764, 8715626185644916, 125200337417147932, 1853095248414187796, 28225529312569364732, 441925530173009732532

Offset: 0

Ilya Gutkovskiy, Dec 05 2023

nonn

Cf. A005493, A035009, A078945, A355247, A367889, A367937.

nmax = 20; CoefficientList[Series[Exp[4 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1) + 2*x))) \\ Michel Marcus, Dec 07 2023

G.f. A(x) satisfies: A(x) = 1 + 2 * x * ( A(x) + 2 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k+2)^n / k!.
a(0) = 1; a(n) = 2 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

A367937 Expansion of e.g.f. exp(4(exp(x) - 1) + 3x).

Original entry on oeis.org

1, 7, 53, 431, 3741, 34471, 335621, 3438943, 36954285, 415187415, 4864054165, 59278367247, 749926582717, 9829744447495, 133267495918885, 1865916660838847, 26942271261464525, 400673643394972983, 6129834703935247285, 96368617886967750767, 1555302323744129219293

Offset: 0

Ilya Gutkovskiy, Dec 05 2023

nonn

Cf. A005494, A035009, A078945, A355252, A366199, A367889.

nmax = 20; CoefficientList[Series[Exp[4 (Exp[x] - 1) + 3 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1) + 3*x))) \\ Michel Marcus, Dec 07 2023

G.f. A(x) satisfies: A(x) = 1 + x * ( 3 * A(x) + 4 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k+3)^n / k!.
a(0) = 1; a(n) = 3 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

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A355247 Expansion of e.g.f. exp(2*(exp(x) - 1 + x)).

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A335980 Expansion of e.g.f. exp(2 * (1 - exp(-x)) + x).

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A355252 Expansion of e.g.f. exp(2*(exp(x) - 1) + 3*x).

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A367889 Expansion of e.g.f. exp(3*(exp(x) - 1) + 2*x).

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A029706 Sum C(n,k)*b(k), k=1..n, where b(k) is given by A001861.

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A129340 Triangular array read by rows: for n, k >= 1, a(n+1, 1) = 2*a(n, n); a(n+1, k+1) = a(n, k)+a(n+1, k).

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A324133 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 12 and t = 12.

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A355252 Expansion of e.g.f. exp(2(exp(x) - 1) + 3x).

A367889 Expansion of e.g.f. exp(3(exp(x) - 1) + 2x).

A366199 Expansion of e.g.f. exp(4(exp(x) - 1) + 2x).

A367937 Expansion of e.g.f. exp(4(exp(x) - 1) + 3x).