A032443 -id:A032443 - cp's OEIS frontend

A356281 a(n) = Sum_{k=0..n} binomial(2n, n-k) q(k), where q(k) is the number of partitions into distinct parts (A000009).

1, 3, 11, 43, 172, 695, 2823, 11501, 46940, 191791, 784148, 3207196, 13119733, 53670793, 219545353, 897957702, 3672093558, 15013596535, 61370565546, 250803861369, 1024716136043, 4185683293934, 17093143284723, 69786349712519, 284847779542644, 1162385753008079

Offset: 0

Vaclav Kotesovec, Aug 01 2022

nonn

Cf. A000009, A032443, A266232, A307496, A356268, A356280.

Table[Sum[PartitionsQ[k]*Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k]*((1-2*x-Sqrt[1-4*x])/(2*x))^k / Sqrt[1-4*x], {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^(2*n - 1/2) * exp(3^(1/3) * Pi^(4/3) * n^(1/3) / 2^(8/3)) / sqrt(3*Pi*n).

A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2n,n-3k).

Original entry on oeis.org

1, 2, 6, 21, 78, 297, 1145, 4447, 17358, 68001, 267141, 1051767, 4148281, 16385111, 64797543, 256515731, 1016368078, 4030114641, 15990813773, 63485616391, 252175202373, 1002136689071, 3984080489263, 15844839393411, 63036297959993, 250855287692647

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A105872, A144904, A360150, A360151, A360152, A360153.
Cf. A000108, A032443, A114121.
Cf. A002450, A371777.

A360168 := proc(n)
    add(binomial(2*n,n-3*k),k=0..n/3) ;
end proc:
seq(A360168(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a[n_] := Sum[Binomial[2*n, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)

a(n) = sum(k=0, n\3, binomial(2*n, n-3*k));

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^6)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^6) ), where c(x) is the g.f. of A000108.
D-finite with recurrence n*a(n) +2*(-4*n+3)*a(n-1) +8*(2*n-3)*a(n-2) +3*(-n+2)=0. - R. J. Mathar, Mar 12 2023
a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(n-2)). - Seiichi Manyama, Apr 10 2024

A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 19 2021

nonn

First differs from A096825 at a(525) = 3, A096825(525) = 4.
First differs from A345926 at a(90) = 4, A345926(90) = 3.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime factors is also the reverse-alternating sum of reversed prime factors.
Also the number of distinct "sums of prime factors" of divisors d|n such that bigomega(d) = bigomega(n)/2 rounded up.

			The divisors of 525 with 2 prime factors are: 15, 21, 25, 35, with prime factors {3,5}, {3,7}, {5,5}, {5,7}, with distinct sums {8,10,12}, so a(525) = 3.

The half-length submultisets are counted by A114921.
Including all multisets of prime factors gives A305611(n) + 1.
The strict rounded version appears to be counted by A342343.
The version for prime indices instead of prime factors is A345926.
A000005 counts divisors, which add up to A000203.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices, row sums of A112798.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A276024 and A299701 count positive subset-sums of partitions.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A334968 counts subsequence-sums of standard compositions.
Cf. A008549, A032443, A083399, A096825, A344609, A345957.

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Length[Union[Total/@Subsets[prifac[n],{Ceiling[PrimeOmega[n]/2]}]]],{n,100}]

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A343943(n):
    fs = factorint(n)
    return len(set(sum(d) for d in multiset_combinations(fs,(sum(fs.values())+1)//2))) # Chai Wah Wu, Aug 23 2021

A356269 a(n) = Sum_{k=0..n} binomial(2k, k) p(k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 3, 15, 75, 425, 2189, 12353, 63833, 346973, 1805573, 9565325, 49069517, 257289529, 1307750129, 6723491129, 34024174649, 172873744739, 865954792079, 4359881882579, 21679061144579, 108108834714719, 534409071271199, 2642716232918639, 12975671796056639, 63765647596939139

Offset: 0

Vaclav Kotesovec, Aug 01 2022

nonn

Cf. A000041, A006134, A032443, A218481, A286955, A356267, A356270.

Table[Sum[Binomial[2*k, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]

a(n) ~ binomial(2*n,n) * p(n) * 4/3.

a(n) ~ 2^(2*n) * exp(Pi*sqrt(2*n/3)) / (3^(3/2) * sqrt(Pi) * n^(3/2)).

A356270 a(n) = Sum_{k=0..n} binomial(2k, k) q(k), where q(k) is the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 3, 9, 49, 189, 945, 4641, 21801, 99021, 487981, 2335541, 10800725, 51363065, 238573865, 1121139065, 5309312105, 24543884585, 113220920945, 530677144745, 2439321389945, 11261499234425, 52169097691865, 239433905462945, 1095710701133345, 5029918350471545

Offset: 0

Vaclav Kotesovec, Aug 01 2022

nonn

Cf. A000009, A006134, A032443, A266232, A307496, A356268, A356269.

Table[Sum[Binomial[2*k, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]

a(n) ~ binomial(2*n,n) * q(n) * 4/3.

a(n) ~ 2^(2*n) * exp(Pi*sqrt(n/3)) / (3^(5/4) * sqrt(Pi) * n^(5/4)).

A360144 a(n) = Sum_{k=0..n} binomial(2n+3k,n-k).

Original entry on oeis.org

1, 3, 14, 69, 344, 1721, 8621, 43206, 216570, 1085574, 5441294, 27272044, 136679882, 684959516, 3432431414, 17199626276, 86182614207, 431824008713, 2163629549132, 10840520569183, 54313805146415, 272122594209738, 1363372115057995, 6830627007245263

Offset: 0

Seiichi Manyama, Jan 27 2023

nonn

Cf. A001700, A032443, A108080, A360143.

Cf. A000108, A000344, A000984.

A360144 := proc(n)
    add(binomial(2*n+3*k,n-k),k=0..n) ;
end proc:
seq(A360144(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a(n) = sum(k=0, n, binomial(2*n+3*k, n-k));

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^5)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^5) ), where c(x) is the g.f. of A000108.
D-finite with recurrence +n*(697*n-7543)*a(n) +(697*n^2+23641*n-3800)*a(n-1) +2*(-32006*n^2+199879*n-255053)*a(n-2) +(283953*n^2-2288641*n+4072186)*a(n-3) +2*(-186566*n^2+1774989*n-4013515)*a(n-4) +(146221*n^2-1648033*n+4472550)*a(n-5) +(38223*n^2-307771*n+532906)*a(n-6) -10*(1511*n-6875)*(2*n-13)*a(n-7)=0. - R. J. Mathar, Mar 12 2023
a(n) = binomial(2*n, n)*hypergeom([1, (1+2*n)/3, 2*(1+n)/3, 1+2*n/3, -n], [(1+n)/4, (2+n)/4, (3+n)/4, 1+n/4], -3^3/4^4). - Stefano Spezia, Jun 17 2025

A386960 a(n) = Sum_{k=0..n} 8^k * binomial(2*n,n-k).

Original entry on oeis.org

1, 10, 102, 1036, 10502, 106380, 1077276, 10908096, 110447046, 1118286172, 11322685172, 114642332232, 1160754172316, 11752638152824, 118995469654968, 1204829162684136, 12198895398209862, 123513816397462524, 1250577392936568708, 12662096110945862856, 128203723152486704052

Offset: 0

Seiichi Manyama, Aug 11 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A032443, A072547, A088218, A100192, A100193.

[&+[8^k * Binomial(2*n, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 13 2025

Table[Sum[8^k*Binomial[2*n,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 13 2025 *)

a(n) = sum(k=0, n, 8^k*binomial(2*n, n-k));

a(n) = [x^n] 1/((1-9*x) * (1-x)^n).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} 9^k * binomial(2*n-k-1,n-k).
G.f.: (1+sqrt(1-4*x))/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7) ).

A113955 Riordan array (1/((1-4x)c(x)),xc(x)/sqrt(1-4x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 3, 1, 11, 6, 1, 42, 30, 9, 1, 163, 140, 58, 12, 1, 638, 630, 325, 95, 15, 1, 2510, 2772, 1686, 624, 141, 18, 1, 9908, 12012, 8330, 3682, 1064, 196, 21, 1, 39203, 51480, 39796, 20264, 7050, 1672, 260, 24, 1, 155382, 218790, 185517, 106203, 42849, 12303, 2475

Offset: 0

Paul Barry, Nov 09 2005

Columns include A032443,A002457,A018218,A038836. Row sums are A100192. Diagonal sums are A113956.

			Triangle begins
1;
3, 1;
11, 6, 1;
42, 30, 9, 1;
163, 140, 58, 12, 1;
638, 630, 325, 95, 15, 1;

Riordan array ((1/(1-4x)+1/sqrt(1-4x))/2, (2x/((1-4x)+sqrt(1-4x)))); Number triangle T(n, k)=sum{j=0..n, C(j, j-k)C(2n, n-j)}.

T(n,k)=sum{j=0..n, C(2n,j)C(n-j,k)}; - Paul Barry, Apr 03 2006

A124834 Triangle, read by rows, where the g.f. of column k, C_k(x), is equal to the product: C_k(x) = Product_{k=0..n} 1/(1 - binomial(n,k)*x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 8, 1, 1, 5, 26, 42, 16, 1, 1, 6, 57, 184, 163, 32, 1, 1, 7, 120, 731, 1358, 638, 64, 1, 1, 8, 247, 2736, 10121, 10244, 2510, 128, 1, 1, 9, 502, 9844, 70436, 145475, 78320, 9908, 256, 1, 1, 10, 1013, 34448, 468735, 1911956, 2141835

Offset: 0

Paul D. Hanna, Nov 09 2006

			Column g.f.s begin:
C_0(x) = 1/(1-x);
C_1(x) = 1/((1-x)(1-x));
C_2(x) = 1/((1-x)(1-2x)(1-x));
C_3(x) = 1/((1-x)(1-3x)(1-3x)(1-x));
C_4(x) = 1/((1-x)(1-4x)(1-6x)(1-4x)(1-x)); ...
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 4, 1;
1, 4, 11, 8, 1;
1, 5, 26, 42, 16, 1;
1, 6, 57, 184, 163, 32, 1;
1, 7, 120, 731, 1358, 638, 64, 1;
1, 8, 247, 2736, 10121, 10244, 2510, 128, 1;
1, 9, 502, 9844, 70436, 145475, 78320, 9908, 256, 1;
1, 10, 1013, 34448, 468735, 1911956, 2141835, 604160, 39203, 512, 1; ...

Cf. A124835 (row sums), A124836 (central terms).

{T(n,k)=polcoeff(1/prod(j=0,k,1-binomial(k,j)*x +x*O(x^n)),n-k)}

T(n+1,n) = 2^n. T(n+2,n) = A032443(n) = Sum_{i=0..n} binomial(2*n,i).

A203578 Exponential (or binomial) half-convolution of A000045 (Fibonacci) with itself.

Original entry on oeis.org

0, 0, 2, 3, 14, 35, 155, 371, 1518, 3891, 15745, 40755, 161459, 426803, 1671175, 4469555, 17301630, 46805811, 179569163, 490156851, 1865624365, 5132989235, 19404565567, 53753361203, 201986220339, 562912506675, 2103942223775, 5894896300851, 21927151270703, 61732155503411

Offset: 0

Wolfdieter Lang, Jan 13 2012

For the definition of the exponential (also known as binomial) half-convolution of a sequence with itself see a comment on A203576 where also the rule for the e.g.f. is given.

Michael De Vlieger, Table of n, a(n) for n = 0..1962
Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.

Cf. A000045, 2*A014335 (exponential convolution), A032443.

Table[Sum[Binomial[n,k]Fibonacci[k]Fibonacci[n-k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Mar 04 2013 *)

a(n) = sum(binomial(n,k)*F(k)*F(n-k),k=0..floor(n/2)), n>=0, with F(n)=A000045(n).
E.g.f.: (f(x)^2 + Fs2(x^2))/2, with the e.g.f. f(x) of A000045 and the o.g.f. Fs2(x):=sum((F(n)/n!)^2*x^n,n=0..infty) of the scaled squares. f(x)^2 = 2*exp(x)*(cosh((2*phi-1)*x)-1)/5 (see A000045 for f(x)) and  Fs2(x^2) = (BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) - 2*BesselI(0,2*i*x))/5, with the golden section phi:=(1+sqrt(5))/2, the complex unit i, and for BesselI see Abramowitz-Stegun (reference and link given in A008277, p. 375, eq. 9.6.10). BesselI(0,2*sqrt(y)) = hypergeom([],[1],y) is the e.g.f. of the sequence {1/n!}.
Bisection:
a(2*k) = (A032443(k)*L(2*k) - (1 + (-1)^k*binomial(2*k,k)))/5 and a(2*k) = (2^(2*k)*L(2*k+1) - 1)/5, k>=0, with the Lucas numbers L(n)=A000032(n), and A032443(k)=(2^(2*k) + binomial(2*k,k))/2. - Wolfdieter Lang, Jan 16 2012.

cp's OEIS Frontend

A356281 a(n) = Sum_{k=0..n} binomial(2*n, n-k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

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A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,n-3*k).

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A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

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A356269 a(n) = Sum_{k=0..n} binomial(2*k, k) * p(k), where p(k) is the partition function A000041.

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A356270 a(n) = Sum_{k=0..n} binomial(2*k, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

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A360144 a(n) = Sum_{k=0..n} binomial(2*n+3*k,n-k).

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Maple

PARI

PARI

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A386960 a(n) = Sum_{k=0..n} 8^k * binomial(2*n,n-k).

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Magma

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A113955 Riordan array (1/((1-4x)c(x)),xc(x)/sqrt(1-4x)), c(x) the g.f. of A000108.

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A124834 Triangle, read by rows, where the g.f. of column k, C_k(x), is equal to the product: C_k(x) = Product_{k=0..n} 1/(1 - binomial(n,k)*x).

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A356281 a(n) = Sum_{k=0..n} binomial(2n, n-k) q(k), where q(k) is the number of partitions into distinct parts (A000009).

A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2n,n-3k).

A356269 a(n) = Sum_{k=0..n} binomial(2k, k) p(k), where p(k) is the partition function A000041.

A356270 a(n) = Sum_{k=0..n} binomial(2k, k) q(k), where q(k) is the number of partitions into distinct parts (A000009).

A360144 a(n) = Sum_{k=0..n} binomial(2n+3k,n-k).