A061256 -id:A061256 - cp's OEIS frontend

A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2n.

10, 22, 48, 104, 224, 480, 1024, 2176, 4608, 9728, 20480, 43008, 90112, 188416, 393216, 819200, 1703936, 3538944, 7340032, 15204352, 31457280, 65011712, 134217728, 276824064, 570425344, 1174405120, 2415919104, 4966055936

Offset: 8

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

This sequence is part of a family of sequences, namely R(n,k), the number of k's in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862. General formula: R(n,k) = 2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k) = 2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2*k.

			a(8)=10 since the palindromic compositions of 15 that contain a 7 are 7+1+7, 4+7+4, 1+3+7+3+1, 3+1+7+1+3, 2+2+7+2+2, 1+1+1+1+7+1+1+1+1, 1+1+2+7+2+1+1, 1+2+1+7+1+2+1 and 2+1+1+7+1+1+2, for a total of 10 7's.

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The Frequency of Summands of a Particular Size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A079859, A061256, A079862, A079863, A111297.

[(2+n)*2^(n-8) : n in [8..40]]; // Vincenzo Librandi, Sep 22 2011

Table[(2 + i)*2^(i - 8), {i, 8, 50}]
LinearRecurrence[{4,-4},{10,22},50] (* Harvey P. Dale, Jun 04 2025 *)

Vec(-2*x^8*(9*x-5)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Dec 16 2014

a(n) = (2+n)*2^(n-8).
a(n) = 2*A111297(n-6). - Colin Barker, Dec 16 2014
a(n) = 4*a(n-1) - 4*a(n-2). - Colin Barker, Dec 16 2014
G.f.: -2*x^8*(9*x-5) / (2*x-1)^2. - Colin Barker, Dec 16 2014
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=8} 1/a(n) = 1024*log(2) - 447047/630.
Sum_{n>=8} (-1)^n/a(n) = 261617/630 - 1024*log(3/2). (End)

A294361 E.g.f.: exp(Sum_{n>=1} sigma(n) * x^n).

Original entry on oeis.org

1, 1, 7, 43, 409, 3841, 50431, 648187, 10347793, 170363809, 3200390551, 62855417131, 1371594161257, 31147757782753, 768384638386639, 19814802390611131, 545309251861956001, 15661899520801953217, 475833949719419469223, 15042718034104688144299

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 14 2017: (Start)
The terms of the sequence appear to be of the form 6*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(7*n+2) == 0 (mod 7); a(11*n+9) == 0 (mod 11); a(13*n+11) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..430
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): A294363 (k=0), this sequence (k=1), A294362 (k=2).

Cf. A000203, A053529, A274804, A061256, A192065.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 04 2018 *)

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

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000203(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(k*x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
a(n) ~ Pi^(1/3) * exp((3*Pi)^(2/3) * n^(2/3)/2 - 3^(1/3) * n^(1/3) / (2*Pi^(2/3)) + 1/24 - 1/(8*Pi^2) - n) * n^(n - 1/6) / 3^(2/3). - Vaclav Kotesovec, Sep 04 2018

A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, -1, -1, 0, 1, 0, -1, -1, -1, 1, 3, 3, -2, -5, -4, 0, 7, 7, 0, -9, -10, 2, 15, 15, -3, -27, -30, 3, 46, 51, 1, -71, -91, -7, 117, 157, 23, -194, -265, -57, 318, 465, 111, -536, -821, -230, 893, 1456, 505, -1485, -2559, -1036, 2433, 4483, 2022

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320777, A320778, A320779, A320780, A320781, A320782.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-NumberOfPrimeFactors(n))):
seq(a(n), n = 0..59); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[PrimeOmega,100]]

A320777 Inverse Euler transform of the number of distinct prime factors (without multiplicity) function A001221.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1, -1, -2, 1, 3, 1, -2, -2, 1, 0, -4, 0, 6, 6, -4, -8, 1, 4, -4, -5, 10, 16, -4, -25, -7, 17, 5, -16, 2, 42, 12, -58, -48, 40, 59, -27, -44, 67, 86, -103, -187, 36, 236, 45, -213, -5, 284, -23, -526, -188, 663, 520

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320778, A320779, A320780, A320781, A320782.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[PrimeNu,100]]

A320782 Inverse Euler transform of the unsigned Moebius function A008966.

Original entry on oeis.org

1, 1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -2, 3, 0, -1, -3, 6, -3, 0, -6, 12, -6, 0, -9, 23, -17, 0, -15, 47, -40, 8, -24, 91, -101, 34, -46, 181, -230, 109, -92, 354, -534, 323, -208, 690, -1177, 883, -520, 1365, -2603, 2297, -1377, 2760, -5641, 5789, -3721, 5741

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

OEIS Wiki, Euler transform

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Table[Abs[MoebiusMu[n]],{n,30}]]

A288385 Expansion of Product_{k>=1} (1 - x^k)^sigma(k).

Original entry on oeis.org

1, -1, -3, -1, 0, 10, 8, 12, 1, -28, -29, -67, -51, -28, 79, 163, 256, 343, 273, 136, -351, -649, -1446, -1751, -1889, -1453, -124, 1924, 5138, 7608, 10636, 10903, 10054, 3143, -5799, -20521, -37217, -53057, -65661, -66086, -54430, -15648, 37179, 122732

Offset: 0

Seiichi Manyama, Jun 08 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001001, A061256.

Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), this sequence (m=1), A288389 (m=2), A288392 (m=3).

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jun 08 2017

b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*DivisorSigma[1, d], {d,
     Divisors[j]}]*b[n - j], {j, 1, n}]/n];
a[n_] := a[n] = If[n == 0, 1, -Sum[b[n - i]*a[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

Convolution inverse of A061256.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A001001(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 29 2018

A072169 Commuting permutations: number of ordered triples of permutations f, g, h in Symm(n) which all commute.

Original entry on oeis.org

1, 1, 8, 48, 504, 4680, 66240, 856800, 14515200, 242040960, 4775500800, 95520902400, 2175146265600, 50438868480000, 1292330988748800, 34092378448128000, 971277752180736000, 28566680100102144000, 896191466580393984000, 29029508406664077312000

Offset: 0

N. J. A. Sloane, Sep 06 2003. More terms from A061256 from N. J. A. Sloane, Jun 13 2012

nonn

a(1)-a(7) computed by John McKay, Sep 06 2003.

T. D. Noe, Table of n, a(n) for n = 0..200

Column k=3 of A362827.

Cf. A061256, A053529.

for n in {1 .. 5} do G := SymmetricGroup(n); t1 := 0; for g in G do for h in G do for i in G do if g*h eq h*g and g*i eq i*g and h*i eq i*h then t1 := t1+1; end if; end for; end for; end for; n, t1; end for;

nn = 20; b = Table[DivisorSigma[1, n], {n, nn}]; Range[0, nn]! CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}],  x] (* T. D. Noe, Jun 19 2012 *)

Equals A061256(n)*n!.

A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 1, 5, 23, 179, 1279, 13699, 135085, 1764377, 22527521, 344625461, 5283739471, 94562354875, 1685808248383, 33947023942259, 694786150879829, 15613612524749489, 357353282848083265, 8880505496901812197, 224851013929747732231, 6106205671049245677251, 169523515381173773551871

Offset: 0

Ilya Gutkovskiy, May 26 2018

nonn

a(n)/n! is the Euler transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Vaclav Kotesovec, Table of n, a(n) for n = 0..438
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms

Cf. A000203, A007429, A017665, A017666, A028342, A053529, A061256, A072169, A318769, A318814.

with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # Vaclav Kotesovec, Sep 04 2018

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 21; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A017665(k)/A017666(k)).
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k*(1 - x^(j*k)))).
log(a(n)/n!) ~ sqrt(n) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018

A318784 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

Original entry on oeis.org

1, 0, 1, 1, 4, 2, 11, 6, 25, 20, 56, 44, 139, 107, 283, 266, 619, 567, 1317, 1242, 2680, 2705, 5403, 5539, 10947, 11339, 21291, 23013, 41494, 45213, 79991, 88312, 151546, 170908, 284901, 324421, 532505, 611227, 981002, 1142000, 1797451, 2105773, 3268765, 3855050, 5889704, 7004451

Offset: 0

Ilya Gutkovskiy, Sep 03 2018

nonn

Convolution of A061256 and A073592.

Euler transform of A001065.

N. J. A. Sloane, Transforms

Cf. A000203, A001065, A001157, A061256, A073592, A318783.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      (sigma(d)-d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 03 2018

nmax = 45; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^(2 k)/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A001065(k).
G.f.: exp(Sum_{k>=1} sigma_2(k)*x^(2*k)/(k*(1 - x^k))), where sigma_2(k) = sum of squares of divisors of k (A001157).
a(n) ~ exp(3^(2/3) * c^(1/3) * n^(2/3)/2 - Pi^2 * n^(1/3) / (4 * 3^(2/3) * c^(1/3)) - Pi^4/(288*c) - 1/8) * A^(3/2) * c^(1/8) / (3^(5/8) * (2*Pi)^(11/24) * n^(5/8)), where c = (Pi^2 - 6)*Zeta(3) and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Sep 03 2018

A226313 Number of commuting 4-tuples of elements from S_n, divided by n!.

Original entry on oeis.org

1, 8, 21, 84, 206, 717, 1810, 5462, 13859, 38497, 96113, 253206, 620480, 1566292, 3770933, 9212041, 21768608, 51795427, 120279052, 279849177, 639379257, 1459282932, 3283758256, 7369471795, 16351101855, 36147590987, 79162129897, 172646751524, 373527250619, 804631686843, 1721283389932, 3666041417241

Offset: 1

N. J. A. Sloane, Jun 08 2013

nonn

Euler transform of A001001.

Alois P. Heinz, Table of n, a(n) for n = 1..9000
Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830, 2013

Column k=4 of A362826.

Cf. A001001, A061256, A301777.

with(numtheory):
b:= proc(n) option remember; add(d*sigma(d), d=divisors(n)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 06 2015

b[n_] := b[n] = DivisorSum[n, #*DivisorSigma[1, #]&];
a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[j, #*b[#]&]*a[n-j], {j, 1, n}] /n];
Array[a, 40] (* Jean-François Alcover, Mar 27 2017, after Alois P. Heinz *)
nmax = 40; Rest[CoefficientList[Series[Exp[Sum[Sum[Sum[d*DivisorSigma[1, d], {d, Divisors[k]}] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Mar 31 2018 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n,i,1)); for(k=1, 2, v=dirmul(v, vector(n,i,i^k))); EulerT(v)} \\ Andrew Howroyd, May 09 2023

a(n) ~ exp(2^(7/4) * Pi^(3/2) * Zeta(3)^(1/4) * n^(3/4) / (3^(3/2) * 5^(1/4)) - sqrt(5*Zeta(3)*n) / (2^(3/2)*Pi) + (sqrt(Pi) * 5^(1/4) / (2^(15/4) * 3^(3/2) * Zeta(3)^(1/4)) - sqrt(3) * 5^(5/4) * Zeta(3)^(3/4) / (2^(15/4) * Pi^(7/2))) * n^(1/4) - 25*Zeta(3) / (16*Pi^6) + (5 - 2*Zeta(3)) / (192*Pi^2)) * Pi^(1/4) * Zeta(3)^(1/8) / (2^(13/8) * 3^(1/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 26 2018

cp's OEIS Frontend

A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2*n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2*n.

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A294361 E.g.f.: exp(Sum_{n>=1} sigma(n) * x^n).

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A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

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A320777 Inverse Euler transform of the number of distinct prime factors (without multiplicity) function A001221.

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A320782 Inverse Euler transform of the unsigned Moebius function A008966.

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A288385 Expansion of Product_{k>=1} (1 - x^k)^sigma(k).

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A072169 Commuting permutations: number of ordered triples of permutations f, g, h in Symm(n) which all commute.

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A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

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A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2n.