A307602 -id:A307602 - cp's OEIS frontend

A126348 Limit of reversed rows of triangle A126347, in which row sums equal Bell numbers (A000110).

1, 1, 2, 4, 7, 12, 20, 33, 53, 84, 131, 202, 308, 465, 695, 1030, 1514, 2209, 3201, 4609, 6596, 9386, 13284, 18705, 26211, 36561, 50776, 70226, 96742, 132765, 181540, 247369, 335940, 454756, 613689, 825698, 1107755, 1482038, 1977465, 2631664

Offset: 0

Paul D. Hanna, Dec 31 2006

nonn

In triangle A126347, row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

Row sums of A253830. a(n) equals the number of colored compositions of n, as defined in A253830, whose associated color partition has distinct parts. An example is given below. - Peter Bala, Jan 20 2015

			a(5) = 12: The colored compositions (defined in A253830) of 5 whose color partitions have distinct parts are
5(c1), 5(c2), 5(c3), 5(c4), 5(c5),
1(c1) + 4(c2), 1(c1) + 4(c3), 1(c1) + 4(c4),
3(c1) + 2(c2),
2(c1) + 3(c2), 2(c1) + 3(c3), 2(c2) + 3(c3). - _Peter Bala_, Jan 20 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..500 from Seiichi Manyama)

Cf. A126347, A126349; factorial variant: A126471. A253830, A307599, A307601, A307602.

nmax = 50; CoefficientList[Series[Product[(1 - x + x^k)/(1 - x), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2019 *)

{B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
{a(n)=Vec(B(n+1,'q)+O('q^(n*(n-1)/2+1)))[n*(n-1)/2+1]}

{a(n) = local(t); if( n<0, 0, t = 1; polcoeff( sum(k=1, (sqrtint(8*n + 1) - 1)\2, t = t * x^k / (1 - x) / (1 - x^k) + x * O(x^n), 1), n))} /* Michael Somos, Aug 17 2008 */

1 + Sum_{k>0} x^(k * (k + 1) / 2) / ((1 - x)^k * (1 - x) * (1 - x^2) ... (1 - x^k)). - Michael Somos, Aug 17 2008
G.f.: Product_{k>0} (1+x^k/(1-x)). - Vladeta Jovovic, Oct 05 2008
G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1)/(d*(1 - x)^d)). - Ilya Gutkovskiy, Apr 19 2019

A307599 Expansion of Product_{k>=1} (1 - x^k/(1 - x)).

Original entry on oeis.org

1, -1, -2, -2, -1, 2, 6, 11, 15, 16, 11, -2, -26, -61, -105, -152, -192, -209, -183, -89, 98, 400, 830, 1385, 2035, 2715, 3314, 3668, 3556, 2703, 790, -2521, -7550, -14542, -23591, -34546, -46901, -59670, -71261, -79358, -80830, -71690, -47133, -1684, 70504, 175168, 317232

Offset: 0

Seiichi Manyama, Apr 17 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Convolution inverse of A227682.

Cf. A126348, A307601, A307602.

m = 46; CoefficientList[Series[Product[1 - x^k/(1 - x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k/(1-x)))

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, x^k*sumdiv(k, d, 1/(d*(1-x)^d)))))

G.f.: exp( - Sum_{k>=1} x^k * Sum_{d|k} 1/(d*(1-x)^d)).

A307601 Expansion of Product_{k>=1} (1 - x^k/(1 + x)).

Original entry on oeis.org

1, -1, 0, 0, -1, 2, -4, 7, -11, 16, -21, 26, -30, 33, -33, 28, -14, -13, 59, -131, 238, -390, 598, -873, 1225, -1663, 2194, -2822, 3544, -4347, 5202, -6059, 6838, -7420, 7633, -7238, 5911, -3226, -1365, 8552, -19190, 34320, -55189, 83266, -120254, 168094, -228958

Offset: 0

Seiichi Manyama, Apr 18 2019

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A126348, A307599, A307602.

m = 46; CoefficientList[Series[Product[1 - x^k/(1 + x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k/(1+x)))

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, x^k*sumdiv(k, d, 1/(d*(1+x)^d)))))

G.f.: exp(-Sum_{k>=1} x^k * Sum_{d|k} 1/(d*(1+x)^d)).

A307762 L.g.f.: log(Product_{k>=1} (1 + x^k/(1 + x))) = Sum_{k>=1} a(k)*x^k/k.

Original entry on oeis.org

1, -1, 7, -13, 36, -67, 141, -269, 547, -1076, 2146, -4231, 8399, -16661, 33177, -66109, 131921, -263353, 526054, -1051108, 2100840, -4199614, 8396289, -16788239, 33570311, -67131715, 134250784, -268483361, 536940598, -1073843297, 2147631641, -4295183165, 8590249609, -17180328793

Offset: 1

Seiichi Manyama, Apr 27 2019

sign

			L.g.f.: L(x) = x/1 - x^2/2 + 7*x^3/3 - 13*x^4/4 + 36*x^5/5 - 67*x^6/6 + 141*x^7/7 - 269*x^8/8 + ... .
exp(L(x)) = 1 + x + 2*x^3 - x^4 + 4*x^5 - 2*x^6 + 5*x^7 - x^8 + ... + A307602(n)*x^n + ... .

Cf. A307602, A307674, A307675, A307761.

N=66; x='x+O('x^N); Vec(x*deriv(log(prod(k=1, N, 1+x^k/(1+x)))))

N=66; x='x+O('x^N); Vec(x*deriv(sum(k=1, N, x^k*sumdiv(k, d, (-1)^(d+1)/(d*(1+x)^d)))))

Product {k>=1} (1 + x^k/(1 + x)) = exp(Sum_{k>=1} a(k)*x^k/k).

cp's OEIS Frontend

A126348 Limit of reversed rows of triangle A126347, in which row sums equal Bell numbers (A000110).

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A307599 Expansion of Product_{k>=1} (1 - x^k/(1 - x)).

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A307601 Expansion of Product_{k>=1} (1 - x^k/(1 + x)).

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A307762 L.g.f.: log(Product_{k>=1} (1 + x^k/(1 + x))) = Sum_{k>=1} a(k)*x^k/k.

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