A129921 -id:A129921 - cp's OEIS frontend

A295739 Expansion of e.g.f. exp(Sum_{k>=1} d(k)*x^k/k!), where d(k) is the number of divisors of k (A000005).

1, 1, 3, 9, 36, 158, 802, 4434, 26978, 176637, 1243528, 9316519, 74065506, 621187700, 5480130494, 50662481722, 489552042241, 4931215686119, 51668848043427, 561981734692781, 6333882472789914, 73850048237680936, 889461218944314524, 11051067390893340510

Offset: 0

Ilya Gutkovskiy, Nov 26 2017

nonn

Exponential transform of A000005.

Seiichi Manyama, Table of n, a(n) for n = 0..553
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Exponential Transform

Cf. A000005, A006171, A028342, A038200, A129921, A160399, A274804, A294363.

a:=series(exp(add(tau(k)*x^k/k!,k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] DivisorSigma[0, k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

E.g.f.: exp(Sum_{k>=1} A000005(k)*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A000005(k)*a(n-k).

A340903 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_0(k) * a(n-k).

Original entry on oeis.org

1, 1, 4, 20, 139, 1192, 12318, 148318, 2041754, 31616757, 544005172, 10296204096, 212589150300, 4755177958104, 114545293676588, 2956316416222300, 81386676426000157, 2380590235918735576, 73729207700492304684, 2410324868012471929670, 82944575892433740648996

Offset: 0

Ilya Gutkovskiy, Jan 26 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..413

Cf. A000005, A000670, A129921, A295739, A340904.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] DivisorSigma[0, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[1/(1 - Sum[Sum[x^(i j)/(i j)!, {j, 1, nmax}], {i, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, numdiv(k)*x^k/k!)))) \\ Seiichi Manyama, Mar 29 2022

a(n) = if(n==0, 1, sum(k=1, n, numdiv(k)*binomial(n, k)*a(n-k))); \\ Seiichi Manyama, Mar 29 2022

E.g.f.: 1 / (1 - Sum_{i>=1} Sum_{j>=1} x^(i*j) / (i*j)!).

E.g.f.: 1 / (1 - Sum_{k>=1} sigma_0(k) * x^k/k!). - Seiichi Manyama, Mar 29 2022

A327736 Expansion of 1 / (1 - Sum_{i>=1, j>=0} x^(i*2^j)).

Original entry on oeis.org

1, 1, 3, 6, 16, 35, 85, 195, 465, 1081, 2549, 5962, 14016, 32847, 77119, 180866, 424466, 995753, 2336497, 5481712, 12861904, 30176671, 70802913, 166120289, 389761751, 914476925, 2145596677, 5034105820, 11811287658, 27712248159, 65019931641, 152553127471, 357928110743

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Invert transform of A001511.

Cf. A000041, A001511, A092119, A129921.

nmax = 32; CoefficientList[Series[1/(1 - Sum[x^(2^k)/(1 - x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[IntegerExponent[2 k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

G.f.: 1 / (1 - Sum_{k>=0} x^(2^k) / (1 - x^(2^k))).

a(0) = 1; a(n) = Sum_{k=1..n} A001511(k) * a(n-k).

A300671 Expansion of 1/(1 - Sum_{k>=1} x^prime(k)/(1 - x^prime(k))).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 8, 15, 23, 40, 63, 108, 172, 290, 471, 782, 1280, 2119, 3474, 5741, 9432, 15557, 25590, 42180, 69413, 114371, 188276, 310136, 510637, 841045, 1384883, 2280831, 3755862, 6185457, 10185941, 16774695, 27624215, 45492412, 74916559, 123374127, 203172520, 334587577

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

Invert transform of A001221.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A001221, A013939, A112965, A129921, A180305, A293548, A300672.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*nops(ifactors(i)[2]), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

nmax = 42; CoefficientList[Series[1/(1 - Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[1/(1 - Sum[PrimeNu[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeNu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

G.f.: 1/(1 - Sum_{k>=2} A001221(k)*x^k).

A320019 Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 3, 8, 6, 1, 0, 2, 14, 18, 8, 1, 0, 4, 20, 41, 32, 10, 1, 0, 2, 28, 78, 92, 50, 12, 1, 0, 4, 37, 132, 216, 175, 72, 14, 1, 0, 3, 44, 209, 440, 490, 298, 98, 16, 1, 0, 4, 58, 306, 814, 1172, 972, 469, 128, 18, 1

Offset: 0

Peter Luschny, Oct 03 2018

Column k is the k-fold self-convolution of tau (A000005). - Alois P. Heinz, Feb 01 2021

			Triangle starts:
[0] 1
[1] 0, 1
[2] 0, 2,  1
[3] 0, 2,  4,   1
[4] 0, 3,  8,   6,   1
[5] 0, 2, 14,  18,   8,   1
[6] 0, 4, 20,  41,  32,  10,   1
[7] 0, 2, 28,  78,  92,  50,  12,  1
[8] 0, 4, 37, 132, 216, 175,  72, 14,  1
[9] 0, 3, 44, 209, 440, 490, 298, 98, 16, 1

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-4 give: A000007, A000005, A055507, A191829, A375002.
Row sums are A129921.
T(2n,n) gives A340992.
Cf. A319083.

P := proc(n, x) option remember; if n = 0 then 1 else
x*add(numtheory:-tau(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
seq(lprint([n], Trow(n)), n=0..9);
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[tau](n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 01 2021
# Uses function PMatrix from A357368.
PMatrix(10, NumberTheory:-tau); # Peter Luschny, Oct 19 2022

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, DivisorSigma[0, n]],
     With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)

The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
p(n, x) = x*Sum_{k=0..n-1} tau(n-k)*p(k, x).
Sigma[k](n) computes the sum of the k-th power of positive divisors of n. The recurrence applied with k = 0 gives this triangle, with k = 1 gives A319083.
T(n,k) = [x^n] (Sum_{j>=1} tau(j)*x^j)^k. - Alois P. Heinz, Feb 14 2021

A320650 Expansion of 1/(1 - Sum_{k>=1} x^k/(1 - x^(2*k))).

Original entry on oeis.org

1, 1, 2, 5, 10, 22, 48, 103, 222, 481, 1038, 2241, 4842, 10456, 22582, 48776, 105342, 227514, 491386, 1061281, 2292132, 4950510, 10692006, 23092378, 49874474, 107717891, 232646956, 502466304, 1085216744, 2343829586, 5062156694, 10933145610, 23613191032

Offset: 0

Ilya Gutkovskiy, Oct 18 2018

nonn

Invert transform of A001227.

Robert Israel, Table of n, a(n) for n = 0..2986
N. J. A. Sloane, Transforms

Cf. A001227, A129921, A206303.

a:=series(1/(1-add(x^k/(1-x^(2*k)),k=1..100)),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019

nmax = 32; CoefficientList[Series[1/(1 - Sum[x^k/(1 - x^(2 k)), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[Mod[d, 2], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)/(1 - x^k)).
G.f.: 1/(1 + x * (d/dx) log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1)))).
a(0) = 1; a(n) = Sum_{k=1..n} A001227(k)*a(n-k).

A327739 Expansion of 1 / (1 - Sum_{i>=1} Sum_{j=1..i} x^(i*j)).

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 38, 78, 163, 338, 703, 1458, 3031, 6293, 13073, 27150, 56396, 117130, 243289, 505310, 1049552, 2179938, 4527804, 9404355, 19533126, 40570816, 84266725, 175024267, 363530253, 755062265, 1568285122, 3257371187, 6765649491, 14052439669

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Invert transform of A038548.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A038548, A129921, A182269, A211856.

a:= proc(n) option remember; `if`(n<1, 1, add(a(n-i)*
      ceil(numtheory[sigma][0](i)/2), i=1..n))
    end:
seq(a(n), n=0..34);  # Alois P. Heinz, Sep 23 2019

nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k^2)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Floor[(DivisorSigma[0, k] + 1)/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

G.f.: 1 / (1 - Sum_{k>=1} x^(k^2) / (1 - x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A038548(k) * a(n-k).

A129922 Number of 3-Carlitz compositions of n (or, more generally p-Carlitz compositions, p > 1), i.e., words b_1^{i_1}b_2^{i_2}...b_k^{i_k} such that the b_j's and i_j's are positive integers for which Sum_{j=1..k} i_j * b_j = n and, for all j, i_j < p and if b_j = b_(j+1) then i_j + i_(j+1) is not equal to p.

Original entry on oeis.org

1, 1, 3, 4, 12, 22, 51, 101, 225, 465, 1008, 2111, 4528, 9560, 20402, 43222, 92018, 195256, 415243, 881758, 1874288, 3981318, 8460906, 17975132, 38196045, 81152769, 172436680, 366376845, 778476016, 1654054258, 3514494256, 7467412436, 15866507485, 33712418692, 71630875356, 152198161794

Offset: 0

Pawel Hitczenko (phitczenko(AT)math.drexel.edu), Jun 05 2007

nonn

For p=2, the sequence enumerates Carlitz compositions, A003242.

			a(3)=4 because, for p=3, we can write:
  3^{1},
  1^{1} 2^{1},
  2^{1} 1^{1},
  1^{1} 1^{1} 1^{1}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Sylvie Corteel and Paweł Hitczenko, Generalizations of Carlitz Compositions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.8.

Cf. A129921.

Cf. A003242.

b:= proc(n, i, j) option remember;
     `if`(n=0, 1, add(add(`if`(k=i and m+j=3, 0,
      b(n-k*m, k, m)), m=1..min(2, n/k)), k=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 22 2017

b[n_, i_, j_] := b[n, i, j] = If[n == 0, 1, Sum[Sum[If[k == i && m + j == 3, 0, b[n - k m, k, m]], {m, 1, Min[2, n/k]}], {k, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *)

N = 66;  x = 'x + O('x^N);  p=3;
gf = 1/(1-sum(k=1,N, x^k/(1-x^k)-p*x^(k*p)/(1-x^(k*p))));
Vec(gf)  /* Joerg Arndt, Apr 28 2013 */

G.f.: 1/(1 - Sum_{k>0} (z^k/(1-z^k) - 3*z^(k*3)/(1-z^(k*3)))).

For general p the generating function is 1/(1 - Sum_{k>0}(z^k/(1-z^k) - p*z^(k*p)/(1-z^(k*p)))).

Added more terms, Joerg Arndt, Apr 28 2013

A300672 Expansion of 1/(1 - Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k))).

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 8, 10, 23, 32, 64, 98, 187, 296, 543, 891, 1595, 2660, 4694, 7924, 13854, 23556, 40940, 69939, 121122, 207490, 358517, 615292, 1061635, 1824013, 3144404, 5406257, 9314645, 16021922, 27595176, 47478950, 81757104, 140691461, 242232918, 416890645, 717712748, 1235289624

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

Invert transform of A001222.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A001222, A022559, A112967, A129921, A180305, A293549, A300671.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*numtheory[bigomega](i), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

nmax = 41; CoefficientList[Series[1/(1 - Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[1/(1 - Sum[PrimeOmega[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeOmega[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 41}]

G.f.: 1/(1 - Sum_{k>=2} A001222(k)*x^k).

A304963 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1} x^(ijk)).

Original entry on oeis.org

1, 1, 4, 10, 31, 82, 241, 664, 1898, 5316, 15058, 42374, 119718, 337432, 952373, 2685906, 7578248, 21376331, 60306495, 170120330, 479922212, 1353855927, 3819280961, 10774233218, 30394408336, 85743168417, 241883489742, 682358211402, 1924947591447, 5430317571250, 15319043353639

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A007425.

N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A000005, A007425, A011782, A129921, A174465, A280473, A304964.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j, 3)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018

nmax = 30; CoefficientList[Series[1/(1 - Sum[x^(i j k), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}]), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[1/(1 - Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

G.f.: 1/(1 - Sum_{k>=1} A007425(k)*x^k).

cp's OEIS Frontend

A295739 Expansion of e.g.f. exp(Sum_{k>=1} d(k)*x^k/k!), where d(k) is the number of divisors of k (A000005).

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A340903 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * sigma_0(k) * a(n-k).

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A327736 Expansion of 1 / (1 - Sum_{i>=1, j>=0} x^(i*2^j)).

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A300671 Expansion of 1/(1 - Sum_{k>=1} x^prime(k)/(1 - x^prime(k))).

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A320019 Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.

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A320650 Expansion of 1/(1 - Sum_{k>=1} x^k/(1 - x^(2*k))).

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A327739 Expansion of 1 / (1 - Sum_{i>=1} Sum_{j=1..i} x^(i*j)).

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A304963 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1} x^(ijk)).