A293840 -id:A293840 - cp's OEIS frontend

A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 3, 13, 1, 0, 1, 1, 3, 19, 49, 1, 0, 1, 1, 3, 19, 97, 261, 1, 0, 1, 1, 3, 19, 121, 681, 1531, 1, 0, 1, 1, 3, 19, 121, 921, 5971, 9073, 1, 0, 1, 1, 3, 19, 121, 1041, 8491, 50443, 63393, 1, 0, 1, 1, 3, 19, 121, 1041

Offset: 0

Seiichi Manyama, Oct 26 2017

			Square array A(n,k) begins:
   1, 1,   1,   1,   1, ...
   0, 1,   1,   1,   1, ...
   0, 1,   3,   3,   3, ...
   0, 1,  13,  19,  19, ...
   0, 1,  49,  97, 121, ...
   0, 1, 261, 681, 921, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000007, A000012, A118589, A294251, A294252, A294253.
Rows n=0 gives A000012.
Main diagonal gives A293840.
Cf. A070936, A294212, A294254.

B(j,k) is the coefficient of Product_{i=1..k} (1+x^i).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(A000217(k),n)} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A293839 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n/n).

Original entry on oeis.org

1, 1, 2, 8, 38, 238, 1828, 16096, 160604, 1826684, 23018264, 316422304, 4755059848, 77084268712, 1343682876272, 25097562397952, 498130253334032, 10479084018025744, 233353674153699616, 5470193826634531456, 134766983204541259616, 3482705318091355591136

Offset: 0

Seiichi Manyama, Oct 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..439

Cf. A293840, A293841.

Cf. A215915.

nmax = 20; CoefficientList[Series[E^Sum[PartitionsQ[k]*x^k/k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} A000009(k)*a(n-k)/(n-k)! for n > 0.

A293528 E.g.f.: exp(x * Product_{k>0} (1 + x^k)).

Original entry on oeis.org

1, 1, 3, 13, 97, 741, 7291, 81313, 1027713, 14231017, 220911571, 3730744821, 68096325793, 1339705629133, 28225576881867, 634123159354441, 15127595174135041, 381586517104288593, 10147599723510322723, 283846981316172613597, 8324822922497497733601

Offset: 0

Seiichi Manyama, Oct 11 2017

From Peter Bala, Mar 28 2022: (Start)
The congruence a(n+k) == a(n) (mod k) holds for all n and k.
It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 3, 3, 7, 1, 1, 3, 3, 7, ...], a purely periodic sequence with period 5.
3 divides a(3*n+2); 13 divides a(13*n+3) and a(13*n+5); 19 divides a(19*n+5), a(19*n+12) and a(19*n+14). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..431
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000009, A293527, A293840.

nmax = 25; CoefficientList[Series[E^(x*QPochhammer[-1, x]/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 11 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(x*prod(k=1, N, (1+x^k)))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000009(k-1)*a(n-k)/(n-k)! for n > 0.

A293841 E.g.f.: exp(Sum_{n>=1} nA000009(n)x^n).

Original entry on oeis.org

1, 1, 5, 49, 409, 4841, 66541, 1006825, 17349809, 333948529, 6997459861, 159199648961, 3918175462345, 103227624161689, 2901807752857469, 86684932131301561, 2738566218754961761, 91236821580866560865, 3196113263245038385189

Offset: 0

Seiichi Manyama, Oct 17 2017

nonn

From Peter Bala, Mar 28 2022: (Start)
The congruence a(n+k) == a(n) (mod k) holds for all n and k.
It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 5, 9, 9, 1, 1, 5, 9, 9, ...], a purely periodic sequence with exact period 5.
5 divides a(5*n+2), 7 divides a(7*n+3); 17 divides a(17*n+7), a(17*n+8) and a(17*n+11). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..418
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000009, A293528, A293840.

nmax = 20; CoefficientList[Series[E^Sum[k*PartitionsQ[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k^2*A000009(k)*a(n-k)/(n-k)! for n > 0.

A294260 E.g.f.: exp(Sum_{n>=1} A010815(n)*x^n).

Original entry on oeis.org

1, -1, -1, 5, 1, 79, -689, 2981, -7615, -172801, 3621151, -16469531, -240199871, 2722511375, 51840080111, -1987808959291, 12337235928961, 136594696115071, -1167414675803585, -56631124939839931, -1376838916423621759, 69766591820556094799

Offset: 0

Seiichi Manyama, Oct 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..449

Main diagonal of A294254.

Cf. A293840.

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A010815(k)*a(n-k)/(n-k)! for n > 0.

A294261 E.g.f.: exp(Sum_{n>=1} A081362(n)*x^n).

Original entry on oeis.org

1, -1, 1, -7, 49, -301, 2281, -21211, 260737, -3254329, 41086801, -589336111, 9851907121, -170708882917, 3060177746809, -60544788499651, 1298663388032641, -28777111728560881, 665551703689032097, -16413980708818538839, 428253175770218766001

Offset: 0

Seiichi Manyama, Oct 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..437

Main diagonal of A294289.

Cf. A058892, A293840.

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A081362(k)*a(n-k)/(n-k)! for n > 0.

A300514 Expansion of e.g.f. exp(Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 1, 2, 6, 20, 79, 358, 1791, 9854, 58958, 379716, 2617320, 19197327, 149099827, 1221390172, 10515829901, 94865603724, 894302028718, 8788782784778, 89848652800152, 953666248076772, 10491219933196228, 119429574273909421, 1404835599743325765, 17052591331677804136

Offset: 0

Ilya Gutkovskiy, Mar 07 2018

nonn

Exponential transform of A000009.

			E.g.f.: A(x) = 1 + x/1! + 2*x^2/2! + 6*x^3/3! + 20*x^4/4! + 79*x^5/5! + 358*x^6/6! + 1791*x^7/7! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..550
N. J. A. Sloane, Transforms
Index entries for related partition-counting sequences

Cf. A000009, A293839, A293840, A300511, A300515.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*b(j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 07 2018

nmax = 24; CoefficientList[Series[Exp[Sum[PartitionsQ[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[PartitionsQ[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]

E.g.f.: exp(Sum_{k>=1} A000009(k)*x^k/k!).

A327674 Number of colored compositions of n using all colors of an n-set such that the color patterns for parts i are sorted and have i (distinct) colors (in arbitrary order).

Original entry on oeis.org

1, 1, 3, 19, 121, 1041, 11191, 130663, 1731969, 25778161, 432791371, 7752723771, 151553121193, 3178030999729, 71244609480591, 1716351868658911, 43661944977384961, 1173984102030774753, 33302371396771085779, 991402105480284394531, 30912472614894951462681

Offset: 0

Alois P. Heinz, Sep 21 2019

nonn

Differs from A293840 and from A294253 first at n = 6.

			a(3) = 19: 3abc, 3acb, 3bac, 3bca, 3cab, 3cba, 2ab1c, 2ac1b, 2ba1c, 2bc1a, 2ca1b, 2cb1a, 1a2bc, 1a2cb, 1b2ac, 1b2ca, 1c2ab, 1c2ba, 1a1b1c.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Main diagonal of A327673.

Cf. A293840, A294253.

b:= proc(n, i, k, p) option remember;
     `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))
    end:
a:= n-> add(b(n$2, i, 0)*(-1)^(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..21);

b[n_, i_, k_, p_] := b[n, i, k, p] =
     If[n == 0, p!, If[i < 1, 0, Sum[Binomial[k^i, j]*
     b[n - i j, Min[n - i j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];
a[n_] := Sum[b[n, n, i, 0] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

a(n) = A327673(n,n).

A293908 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000009(j)x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 5, 19, 38, 1, 1, 9, 49, 121, 238, 1, 1, 17, 133, 409, 1041, 1828, 1, 1, 33, 373, 1441, 4841, 10651, 16096, 1, 1, 65, 1069, 5233, 23601, 66541, 121843, 160604, 1, 1, 129, 3109, 19441, 119441, 442681, 1006825, 1575729, 1826684, 1, 1

Offset: 0

Seiichi Manyama, Oct 19 2017

			Square array begins:
     1,    1,    1,     1,      1, ...
     1,    1,    1,     1,      1, ...
     2,    3,    5,     9,     17, ...
     8,   19,   49,   133,    373, ...
    38,  121,  409,  1411,   5233, ...
   238, 1041, 4841, 23601, 119441, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A293839, A293840, A293841.
Rows n=0-1 give A000012.
Cf. A293796.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j^k*A000009(j)*A(n-j,k)/(n-j)! for n > 0.

A320899 Expansion of e.g.f. exp(1/theta_4(x) - 1), where theta_4() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 12, 104, 1120, 14592, 221824, 3835904, 74262528, 1589016320, 37181031424, 943547716608, 25791165349888, 754934109863936, 23547020011929600, 779291847538638848, 27263652732032843776, 1005002283128197349376, 38921431600215853760512, 1579513585265275661189120

Offset: 0

Ilya Gutkovskiy, Oct 23 2018

nonn

Cf. A015128, A058892, A293840.

seq(coeff(series(factorial(n)*(exp(-1+mul((1+x^k)/(1-x^k),k=1..n))),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 23 2018

nmax = 19; CoefficientList[Series[Exp[1/EllipticTheta[4, 0, x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Sum[PartitionsP[k - j] PartitionsQ[j], {j, 0, k}] k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]

E.g.f.: exp(-1 + Product_{k>=1} (1 + x^k)/(1 - x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A015128(k)*k!*binomial(n-1,k-1)*a(n-k).

cp's OEIS Frontend

A294250 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} (1+x^j) - 1).

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A293839 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n/n).

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Mathematica

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A293528 E.g.f.: exp(x * Product_{k>0} (1 + x^k)).

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A293841 E.g.f.: exp(Sum_{n>=1} n*A000009(n)*x^n).

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Mathematica

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A294260 E.g.f.: exp(Sum_{n>=1} A010815(n)*x^n).

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A294261 E.g.f.: exp(Sum_{n>=1} A081362(n)*x^n).

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A300514 Expansion of e.g.f. exp(Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

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A327674 Number of colored compositions of n using all colors of an n-set such that the color patterns for parts i are sorted and have i (distinct) colors (in arbitrary order).

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Maple

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A293908 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)*A000009(j)*x^j).

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A293841 E.g.f.: exp(Sum_{n>=1} nA000009(n)x^n).

A293908 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000009(j)x^j).