A000726 -id:A000726 - cp's OEIS frontend

A091605 Column 2 of triangle A091602.

1, 0, 2, 2, 3, 4, 7, 8, 12, 15, 21, 26, 35, 43, 57, 70, 89, 109, 138, 167, 208, 251, 309, 371, 452, 539, 652, 775, 929, 1099, 1311, 1543, 1829, 2146, 2529, 2957, 3469, 4040, 4721, 5481, 6377, 7381, 8559, 9875, 11412, 13133, 15128, 17364, 19945, 22833

Offset: 2

Christian G. Bower, Jan 23 2004

nonn

			a(7) counts these partitions:  511, 331, 322, 3211. - _Clark Kimberling_, Mar 10 2014

Alois P. Heinz, Table of n, a(n) for n = 2..1000

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 2 &];
t = Table[d[n], {n, 12}]  (* shows partitions *)
u = Table[Length[d[n]], {n, 2, 30}] (* counts partitions *)
(* Clark Kimberling, Mar 10 2014 *)

a(n) = A000726(n) - A000009(n).

A266649 Expansion of Product_{k>=1} 1 - x^(3*k)/(1-x^k).

Original entry on oeis.org

1, 0, 0, -1, -1, -1, -2, -1, -2, -1, -1, 1, -1, 3, 2, 5, 4, 8, 7, 11, 8, 12, 11, 13, 10, 9, 9, 7, 3, -2, -5, -13, -16, -25, -28, -48, -44, -66, -60, -82, -82, -104, -95, -120, -103, -131, -107, -133, -98, -124, -85, -94, -42, -51, 9, 7, 83, 100, 181, 208, 298

Offset: 0

Vaclav Kotesovec, Jan 02 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A100405, A266647, A266648, A266650.

nmax = 100; CoefficientList[Series[Product[1-x^(3*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k - x^(3*k)) / (1 - x^k).

A275820 Expansion of Product_{k>=1} (1 + x^(2k) + x^(3k)).

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 3, 1, 3, 3, 3, 2, 7, 3, 8, 7, 10, 7, 16, 8, 17, 17, 21, 17, 35, 22, 37, 36, 46, 37, 69, 46, 74, 71, 91, 81, 128, 96, 144, 139, 173, 154, 236, 185, 263, 257, 314, 286, 417, 345, 470, 462, 557, 517, 719, 617, 815, 802, 960, 904, 1211, 1068

Offset: 0

Vaclav Kotesovec, Nov 15 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A263401, A264905, A266686, A275821.

nmax = 100; CoefficientList[Series[Product[1+x^(2*k)+x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[3]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + If[j < 2*k, 0, p[[j - 2*k + 1]]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-2*x) + exp(-3*x)) dx = 0.60248650631158778882474716370201988195290074160793967143564...

A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).

Original entry on oeis.org

1, 1, 3, 12, 72, 480, 3780, 35280, 372960, 4263840, 54432000, 758419200, 11436163200, 185253868800, 3214699488000, 59172265152000, 1163830187520000, 24097823253504000, 525794940582912000, 12073276215576576000, 290883846352619520000, 7318777466097377280000

Offset: 0

Seiichi Manyama, Oct 01 2017

nonn

			Let's consider the partitions of n where no positive integer appears more than twice. (See A000726)
For n = 5,
    partition      |                         |
--------------------------------------------------------------------
     5             -> one 5                  -> 1/(1!)       (= 1  )
   = 4 + 1         -> one 4 and one 1        -> 1/(1!*1!)    (= 1  )
   = 3 + 2         -> one 3 and one 2        -> 1/(1!*1!)    (= 1  )
   = 3 + 1 + 1     -> one 3 and two 1        -> 1/(1!*2!)    (= 1/2)
   = 2 + 2 + 1     -> two 2 and one 1        -> 1/(2!*1!)    (= 1/2)
--------------------------------------------------------------------
                                                sum             4
So a(5) = 5! * 4 = 480.
For n = 6,
    partition      |                         |
--------------------------------------------------------------------
     6             -> one 6                  -> 1/(1!)       (= 1  )
   = 5 + 1         -> one 5 and one 1        -> 1/(1!*1!)    (= 1  )
   = 4 + 2         -> one 4 and one 2        -> 1/(1!*1!)    (= 1  )
   = 4 + 1 + 1     -> one 4 and two 1        -> 1/(1!*2!)    (= 1/2)
   = 3 + 3         -> two 3                  -> 1/(2!)       (= 1/2)
   = 3 + 2 + 1     -> one 3, one 2 and one 1 -> 1/(1!*1!*1!) (= 1  )
   = 2 + 2 + 1 + 1 -> two 2 and two 1        -> 1/(2!*2!)    (= 1/4)
--------------------------------------------------------------------
                                                sum            21/4
So a(6) = 6! * 21/4 = 3780.

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=2 of A293135.

Cf. A000726, A162891, A263401, A293141.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)/j!, j=0..min(2, n/i))))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 02 2017

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1]/j!, {j, 0, Min[2, n/i]}]]];
a[n_] := n! b[n, n];
a /@ Range[0, 23] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n) - n) * n^(n+1/2) / (sqrt(5) * n^(3/4)), where c = -polylog(2, -1/2 - I/2) - polylog(2, -1/2 + I/2) = 0.9669456127221570300837545... Equivalently, c = -Sum_{k>=1} (-1)^k * cos(Pi*k/4) / (k^2 * 2^(k/2-1)). - Vaclav Kotesovec, Oct 01 2017

A293204 G.f.: Product_{m>0} (1+x^m+2!x^(2m)).

Original entry on oeis.org

1, 1, 3, 2, 6, 7, 12, 13, 22, 26, 42, 46, 73, 80, 116, 139, 194, 226, 306, 358, 482, 558, 735, 856, 1108, 1300, 1657, 1926, 2426, 2834, 3530, 4110, 5082, 5898, 7234, 8409, 10216, 11860, 14304, 16568, 19891, 22990, 27470, 31670, 37630, 43382, 51274, 58982, 69450

Offset: 0

Seiichi Manyama, Oct 02 2017

nonn

			Let's consider the partitions of n where no positive integer appears more than twice. (See A000726)
For n = 5,
    partition      |                         |
--------------------------------------------------------------
     5             -> one 5                  -> 1!       (= 1)
   = 4 + 1         -> one 4 and one 1        -> 1!*1!    (= 1)
   = 3 + 2         -> one 3 and one 2        -> 1!*1!    (= 1)
   = 3 + 1 + 1     -> one 3 and two 1        -> 1!*2!    (= 2)
   = 2 + 2 + 1     -> two 2 and one 1        -> 2!*1!    (= 2)
--------------------------------------------------------------
                                                a(5)      = 7.
For n = 6,
    partition      |                         |
--------------------------------------------------------------
     6             -> one 6                  -> 1!       (= 1)
   = 5 + 1         -> one 5 and one 1        -> 1!*1!    (= 1)
   = 4 + 2         -> one 4 and one 2        -> 1!*1!    (= 1)
   = 4 + 1 + 1     -> one 4 and two 1        -> 1!*2!    (= 2)
   = 3 + 3         -> two 3                  -> 2!       (= 2)
   = 3 + 2 + 1     -> one 3, one 2 and one 1 -> 1!*1!*1! (= 1)
   = 2 + 2 + 1 + 1 -> two 2 and two 1        -> 2!*2!    (= 4)
--------------------------------------------------------------
                                                a(6)      = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 501 terms from Seiichi Manyama)

Column k=2 of A293202.
Cf. A000726, A263401, A293138, A293182.
Cf. A293072.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*j!, j=0..min(2, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 02 2017

nmax = 100; CoefficientList[Series[Product[1 + x^k + 2*x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 02 2017 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4 * sqrt(Pi) * n^(3/4)), where c = Pi^2/3 - arctan(sqrt(7))^2 + log(2)^2/4 + polylog(2, -1/4 - I*sqrt(7)/4) + polylog(2, -1/4 + I*sqrt(7)/4) = 1.323865936864425754643630663383779192757247984691212163137... - Vaclav Kotesovec, Oct 02 2017

Equivalently, c = -polylog(2, -1/2 + I*sqrt(7)/2) - polylog(2, -1/2 - I*sqrt(7)/2). - Vaclav Kotesovec, Oct 05 2017

A342532 Number of even-length compositions of n with alternating parts distinct.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 9, 14, 28, 44, 83, 136, 250, 424, 757, 1310, 2313, 4018, 7081, 12314, 21650, 37786, 66264, 115802, 202950, 354858, 621525, 1087252, 1903668, 3330882, 5831192, 10204250, 17862232, 31260222, 54716913, 95762576, 167614445, 293356422, 513456686

Offset: 0

Gus Wiseman, Mar 28 2021

nonn

These are finite even-length sequences q of positive integers summing to n such that q(i) != q(i+2) for all possible i.

			The a(2) = 1 through a(7) = 14 compositions:
  (1,1)  (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
         (2,1)  (2,2)  (2,3)  (2,4)      (2,5)
                (3,1)  (3,2)  (3,3)      (3,4)
                       (4,1)  (4,2)      (4,3)
                              (5,1)      (5,2)
                              (1,1,2,2)  (6,1)
                              (1,2,2,1)  (1,1,2,3)
                              (2,1,1,2)  (1,1,3,2)
                              (2,2,1,1)  (1,2,3,1)
                                         (1,3,2,1)
                                         (2,1,1,3)
                                         (2,3,1,1)
                                         (3,1,1,2)
                                         (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..500

The strictly decreasing version appears to be A064428 (odd-length: A001522).
The equal version is A065608 (A342527 with odds).
The weakly decreasing version is A114921 (A342528 with odds).
Including odds gives A224958.
A000726 counts partitions with alternating parts unequal.
A325545 counts compositions with distinct first differences.
A342529 counts compositions with distinct first quotients.
Cf. A000009, A000041, A003242, A008965, A032020, A059966, A062968, A064410, A070211, A106351, A175342, A325546.

qdq[q_]:=And@@Table[q[[i]]!=q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],qdq]],{n,0,15}]

\\ here gf gives A106351 as g.f.
gf(n, y)={1/(1 - sum(k=1, n, (-1)^(k+1)*x^k*y^k/(1-x^k) + O(x*x^n)))}
seq(n)={my(p=gf(n,y)); Vec(sum(k=0, n\2, polcoef(p,k,y)^2))} \\ Andrew Howroyd, Apr 16 2021

G.f.: 1 + Sum_{k>=1} B_k(x)^2 where B_k(x) is the g.f. of column k of A106351. - Andrew Howroyd, Apr 16 2021

Terms a(24) and beyond from Andrew Howroyd, Apr 16 2021

A285927 Expansion of (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^3 in powers of x.

Original entry on oeis.org

1, 3, 9, 19, 42, 81, 155, 276, 486, 821, 1368, 2214, 3541, 5544, 8586, 13082, 19740, 29403, 43414, 63423, 91935, 132075, 188418, 266733, 375232, 524331, 728514, 1006216, 1382604, 1889739, 2570719, 3480420, 4691682, 6297102, 8418252, 11209347, 14870970

Offset: 0

Seiichi Manyama, Apr 28 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Mohammed L. Nadji and Moussa Ahmia, Congruences for L-regular tripartitions for L in {2, 3}, Integers (2024) Vol. 24, Art. No. A86. See p. 2.

(Product_{k>0} (1 - x^(m*k)) / (1 - x^k))^m: A022567 (m=2), this sequence (m=3), A093160 (m=4), A285928 (m=5).

Cf. A000726, A199659.

nmax = 40; CoefficientList[Series[Product[((1 - x^(3*k)) / (1 - x^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 30 2017 *)

a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A046913(k)*a(n-k) for n > 0.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(7/4) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

A327717 Expansion of Product_{k>=1} (1 + x^k/(1 + x^(2*k))).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 3, 5, 6, 6, 7, 10, 12, 12, 15, 20, 23, 24, 28, 36, 42, 44, 51, 64, 73, 78, 89, 108, 123, 132, 150, 179, 202, 218, 246, 288, 324, 350, 393, 456, 509, 552, 616, 706, 786, 852, 948, 1078, 1195, 1297, 1436, 1620, 1791, 1942, 2145, 2406, 2650, 2874, 3163, 3528

Offset: 0

Seiichi Manyama, Sep 23 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Convolution inverse of A307757.

Cf. A000726, A327716, A327718, A327719, A327720.

nmax = 100; CoefficientList[Series[Product[1 + x^k/(1 + x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^k + x^(2*k)) * (1 - x^(4*k - 2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k/(1+x^(2*k))))

a(n) ~ 5^(1/4) * exp(sqrt(5*n/2)*Pi/3) / (2^(5/4)*3*n^(3/4)). - Vaclav Kotesovec, Sep 23 2019

A328547 Number of 3-regular bipartitions of n.

Original entry on oeis.org

1, 2, 5, 8, 16, 26, 44, 68, 108, 162, 245, 356, 521, 740, 1053, 1468, 2045, 2804, 3836, 5184, 6988, 9326, 12409, 16376, 21546, 28154, 36674, 47492, 61317, 78764, 100880, 128628, 163553, 207134, 261630, 329288, 413395, 517316, 645803, 803844, 998282

Offset: 0

N. J. A. Sloane, Oct 19 2019

nonn

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Cf. A000726.

f:=(k,M) -> mul(1-q^(k*j),j=1..M);
LRBP := (L,M) -> (f(L,M)/f(1,M))^2;
S := L -> seriestolist(series(LRBP(L,80),q,60));
S(3);

nmax = 40; CoefficientList[Series[Product[1 + x^j + x^(2*j), {j, 1, nmax}]^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2024 *)

a(n) ~ exp(Pi*sqrt(8*n)/3) / (2^(3/4) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 08 2024

A353502 Numbers with all prime indices and exponents > 2.

Original entry on oeis.org

1, 125, 343, 625, 1331, 2197, 2401, 3125, 4913, 6859, 12167, 14641, 15625, 16807, 24389, 28561, 29791, 42875, 50653, 68921, 78125, 79507, 83521, 103823, 117649, 130321, 148877, 161051, 166375, 205379, 214375, 226981, 274625, 279841, 300125, 300763, 357911

Offset: 1

Gus Wiseman, May 16 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The initial terms together with their prime indices:
       1: {}
     125: {3,3,3}
     343: {4,4,4}
     625: {3,3,3,3}
    1331: {5,5,5}
    2197: {6,6,6}
    2401: {4,4,4,4}
    3125: {3,3,3,3,3}
    4913: {7,7,7}
    6859: {8,8,8}
   12167: {9,9,9}
   14641: {5,5,5,5}
   15625: {3,3,3,3,3,3}
   16807: {4,4,4,4,4}
   24389: {10,10,10}
   28561: {6,6,6,6}
   29791: {11,11,11}
   42875: {3,3,3,4,4,4}

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3000 terms from Amiram Eldar)

The version for only parts is A007310, counted by A008483.
The version for <= 2 instead of > 2 is A018256, # of compositions A137200.
The version for only multiplicities is A036966, counted by A100405.
The version for indices and exponents prime (instead of > 2) is:
- listed by A346068
- counted by A351982
- only exponents: A056166, counted by A055923
- only parts: A076610, counted by A000607
The version for > 1 instead of > 2 is A062739, counted by A339222.
The version for compositions is counted by A353428, see A078012, A353400.
The partitions with these Heinz numbers are counted by A353501.
A000726 counts partitions with multiplicities <= 2, compositions A128695.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A295341 counts partitions with some multiplicity > 2, compositions A335464.
Cf. A000720, A003963, A065483, A114901, A181819, A353503, A353508.

Select[Range[10000],#==1||!MemberQ[FactorInteger[#],{?(#<5&),}|{,?(#<3&)}]&]

Sum_{n>=1} 1/a(n) = Product_{p prime > 3} (1 + 1/(p^2*(p-1))) = (72/95)*A065483 = 1.0154153584... . - Amiram Eldar, May 28 2022

cp's OEIS Frontend

A091605 Column 2 of triangle A091602.

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A266649 Expansion of Product_{k>=1} 1 - x^(3*k)/(1-x^k).

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A275820 Expansion of Product_{k>=1} (1 + x^(2*k) + x^(3*k)).

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A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).

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A293204 G.f.: Product_{m>0} (1+x^m+2!*x^(2*m)).

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A342532 Number of even-length compositions of n with alternating parts distinct.

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A285927 Expansion of (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^3 in powers of x.

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A327717 Expansion of Product_{k>=1} (1 + x^k/(1 + x^(2*k))).

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A275820 Expansion of Product_{k>=1} (1 + x^(2k) + x^(3k)).

A293204 G.f.: Product_{m>0} (1+x^m+2!x^(2m)).