A000712 -id:A000712 - cp's OEIS frontend

A307755 Exponential convolution of partition numbers (A000041) with themselves.

1, 2, 6, 18, 58, 184, 586, 1822, 5618, 16980, 50892, 150064, 439210, 1268924, 3640342, 10337596, 29160638, 81570368, 226795202, 626070664, 1718783084, 4689582366, 12730998988, 34373603158, 92385339242, 247099560046, 658137847408, 1745322097886, 4610549234836, 12131656526628

Offset: 0

Ilya Gutkovskiy, Apr 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A000041, A000712, A218481, A307756.

a:= n-> (p-> add(binomial(n, j)*p(j)*p(n-j), j=0..n))(combinat[numbpart]):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 26 2019

nmax = 29; CoefficientList[Series[Sum[PartitionsP[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] PartitionsP[k] PartitionsP[n - k], {k, 0, n}], {n, 0, 29}]

E.g.f.: (Sum_{k>=0} A000041(k)*x^k/k!)^2.
a(n) = Sum_{k=0..n} binomial(n,k)*A000041(k)*A000041(n-k).
a(n) ~ exp(2*Pi*sqrt(n/3)) * 2^(n-2) / (3*n^2). - Vaclav Kotesovec, May 06 2019

A316143 Expansion of e.g.f. Product_{k>=1} 1 / (1 - (exp(x)-1)^k)^2.

Original entry on oeis.org

1, 2, 12, 92, 912, 10772, 148512, 2328692, 40842912, 791302772, 16767551712, 385382491892, 9542377300512, 253105962752372, 7156766466076512, 214814484529608692, 6819311473596695712, 228212485803422931572, 8028037725386962194912, 296094910181041530831092

Offset: 0

Vaclav Kotesovec, Jun 25 2018

Self-convolution of A167137.

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 2, 5, 1, 2, 6, 0, 2, 5, 1, 2, 6, 0, 2, 5, 1, 2, 6, 0, ...], with a preperiod of length 1 and an apparent period thereafter of 6 = phi(7). - Peter Bala, Mar 03 2023

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A000712, A167137, A316142, A316144.

nmax = 20; CoefficientList[Series[Product[1/(1-(Exp[x]-1)^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

Sum_{k=0..n} binomial(n,k) * A167137(k) * A167137(n-k).

a(n) ~ n! * exp(Pi * sqrt(2*n/(3*log(2))) - Pi^2 * (1 - 1/log(2)) / 12) / (2^(7/4) * 3^(3/4) * n^(5/4) * (log(2))^(n - 1/4)).

A349153 Numbers k such that the k-th composition in standard order has sum equal to twice its reverse-alternating sum.

Original entry on oeis.org

0, 11, 12, 14, 133, 138, 143, 148, 155, 158, 160, 168, 179, 182, 188, 195, 198, 204, 208, 216, 227, 230, 236, 240, 248, 2057, 2066, 2071, 2077, 2084, 2091, 2094, 2101, 2106, 2111, 2120, 2131, 2134, 2140, 2149, 2154, 2159, 2164, 2171, 2174, 2192, 2211, 2214

Offset: 1

Gus Wiseman, Nov 17 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

			The terms and corresponding compositions begin:
    0: ()
   11: (2,1,1)
   12: (1,3)
   14: (1,1,2)
  133: (5,2,1)
  138: (4,2,2)
  143: (4,1,1,1,1)
  148: (3,2,3)
  155: (3,1,2,1,1)
  158: (3,1,1,1,2)
  160: (2,6)
  168: (2,2,4)
  179: (2,1,3,1,1)
  182: (2,1,2,1,2)
  188: (2,1,1,1,3)

These compositions are counted by A262977 up to 0's.
Except for 0, a subset of A345917.
The unreversed version is A348614.
The unreversed negative version is A349154.
The negative version is A349155.
A non-reverse unordered version is A349159, counted by A000712 up to 0's.
An unordered version is A349160, counted by A006330 up to 0's.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A294175.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- Heinz number is given by A333219.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Alternating compositions are ranked by A345167, complement A345168.

stc[n_]:=Differences[ Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,1000],Total[stc[#]]==2*sats[stc[#]]&]

A355386 Position of first appearance of n in A355382, where A355382(m) = number of divisors d of m such that bigomega(d) = omega(m); or a(n) = -1 if n does not appear in A355382.

Original entry on oeis.org

1, 12, 36, 120, 180, 360, 840, 1260, 5400, 27000, 2520, 5040, 6300, 7560, 15120, 12600, 25200

Offset: 1

Gus Wiseman, Jul 02 2022

The first position of -1 appears to be 18, pointed out by Amiram Eldar.
The terms are not always increasing.
The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.

			The terms together with their prime indices begin:
      1: {}
     12: {1,1,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    180: {1,1,2,2,3}
    360: {1,1,1,2,2,3}
    840: {1,1,1,2,3,4}
   1260: {1,1,2,2,3,4}
   5400: {1,1,1,2,2,2,3,3}
  27000: {1,1,1,2,2,2,3,3,3}
   2520: {1,1,1,2,2,3,4}
   5040: {1,1,1,1,2,2,3,4}
   6300: {1,1,2,2,3,3,4}
   7560: {1,1,1,2,2,2,3,4}
  15120: {1,1,1,1,2,2,2,3,4}
The terms together with their divisors satisfying the condition begin:
      1:   1
     12:   4,   6
     36:   4,   6,   9
    120:   8,  12,  20,  30
    180:  12,  18,  20,  30,  45
    360:   8,  12,  18,  20,  30,  45
    840:  24,  40,  56,  60,  84, 140, 210
   1260:  36,  60,  84,  90, 126, 140, 210, 315
   5400:   8,  12,  18,  20,  27,  30,  45,  50,  75
  27000:   8,  12,  18,  20,  27,  30,  45,  50,  75, 125
   2520:  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   5040:  16,  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   6300:  36,  60,  84,  90, 100, 126, 140, 150, 210, 225, 315, 350, 525

These are the positions of first appearances in A355382, which is the version of A181591 without multiplicity.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 counts prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
A355383 counts cmpsbl. pairs of partitions with containment, comps. A355384.
Cf. A000712, A022811, A056239, A071625, A181819, A319910, A339006.

tf=Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,1000}];
Table[Position[tf,n][[1,1]],{n,Select[Union[tf],SubsetQ[tf,Range[#]]&]}]

A000715 Number of partitions of n, with three kinds of 1,2 and 3 and two kinds of 4,5,6,....

Original entry on oeis.org

1, 3, 9, 22, 50, 104, 208, 394, 724, 1286, 2229, 3769, 6253, 10176, 16303, 25723, 40055, 61588, 93647, 140875, 209889, 309846, 453565, 658627, 949310, 1358589, 1931464, 2728547, 3831654, 5350119, 7430158, 10265669, 14113795, 19313168, 26309405, 35685523

Offset: 0

N. J. A. Sloane

nonn

Convolution of A000712 and A001399. - Vaclav Kotesovec, Aug 18 2015

			a(2)=9 because we have 2, 2', 2", 1+1, 1'+1', 1"+1", 1+1', 1+1", 1'+1".

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

g:=1/((1-x)*(1-x^2)*(1-x^3)*product((1-x^k)^2,k=1..40)): gser:=series(g,x=0,40): seq(coeff(gser,x,n),n=0..31); # Emeric Deutsch, Apr 17 2006
# second Maple program
a:= proc(n) a(n):= `if`(n=0, 1, add(add(d*`if`(d<4, 3, 2), d=numtheory [divisors](j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..50); # Alois P. Heinz, Sep 25 2012

nn=25;p=Product[1/(1- x^i)^2,{i,1,nn}];CoefficientList[Series[p /(1-x)/(1-x^2)/(1-x^3),{x,0,nn}],x] (* Geoffrey Critzer, Sep 25 2012 *)

EULER transform of 3, 3, 3, 2, 2, 2, 2, 2, ...
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*Product_{k>=1}(1-x^k)^2). - Emeric Deutsch, Apr 17 2006
a(n) ~ exp(2*Pi*sqrt(n/3)) * n^(1/4) / (8 * 3^(1/4) * Pi^3). - Vaclav Kotesovec, Aug 18 2015

Extended with formula from Christian G. Bower, Apr 15 1998

A100535 Number of partitions of 2*n + 1 into parts of two kinds.

Original entry on oeis.org

2, 10, 36, 110, 300, 752, 1770, 3956, 8470, 17490, 35002, 68150, 129512, 240840, 439190, 786814, 1386930, 2408658, 4126070, 6978730, 11664896, 19283830, 31551450, 51124970, 82088400, 130673928, 206327710, 323275512, 502810130

Offset: 0

N. J. A. Sloane, Nov 27 2004

nonn

			G.f.: 2 + 10*x + 36*x^2 + 110*x^3 + 300*x^4 + 752*x^5 + 1770*x^6 + 3956*x^7 + ...
G.f.: 2*q^11 + 10*q^35 + 36*q^59 + 110*q^83 + 300*q^107 + 752*q^131 + 1770*q^155 + ...
a(1)=10 because we have 3, 3', 21, 2'1, 21', 2'1', 111, 1'11, 1'1'1, 1'1'1'.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000712.

m:=40;
f:= func< x | 2*(&*[ ((1-x^(2*n))^2*(1-x^(8*n))^2)/((1-x^n)^5*(1-x^(4*n))) : n in [1..m+2]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( f(x) )); // G. C. Greubel, Mar 27 2023

with(combinat): A000712:=n->sum(numbpart(k)*numbpart(n-k),k=0..n): seq(A000712(2*n-1),n=1..32); # Emeric Deutsch, Dec 16 2004

a[n_]:= Sum[PartitionsP[k] PartitionsP[2n+1-k], {k,0,2n+1}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 30 2015, adapted from Maple *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 2 * eta(x^2 + A)^2 * eta(x^8 + A)^2 / (eta(x + A)^5 * eta(x^4 + A)), n))} /* Michael Somos, Sep 24 2011 */

{a(n) = local(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( 1 / eta(x + A)^2, n))} /* Michael Somos, Sep 24 2011 */

m=40
def f(x): return 2*product( ((1-x^(2*n))^2*(1-x^(8*n))^2)/((1-x^n)^5*(1-x^(4*n))) for n in range(1,m+2) )
def A100535_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A100535_list(m) # G. C. Greubel, Mar 27 2023

Expansion of q^(-11/24) * 2 * eta(q^2)^2 * eta(q^8)^2 / (eta(q)^5 * eta(q^4)) In powers of q. - Michael Somos, Sep 24 2011

a(n) = A000712(2*n + 1).

More terms from Emeric Deutsch, Dec 16 2004

A158139 Number of nondecreasing integer sequences of length 5 with sum zero and sum of absolute values 2n.

Original entry on oeis.org

1, 4, 8, 17, 23, 44, 54, 85, 107, 150, 178, 247, 281, 366, 422, 527, 591, 734, 808, 975, 1079, 1272, 1388, 1633, 1761, 2036, 2204, 2513, 2695, 3068, 3266, 3677, 3923, 4374, 4638, 5167, 5449, 6022, 6362, 6983, 7343, 8054, 8436, 9199, 9647, 10464, 10936

Offset: 1

R. H. Hardin Mar 13 2009

nonn

a(n) = A000041(n)^2 for n<=2

a(n) = A000041(n)^2 - cumulative A000712(2*n-1-length), 0 <= 2*n-1-length <= floor(n/2) [empirical].

R. H. Hardin, Table of n, a(n) for n=1..999

A158184 Number of nondecreasing integer sequences of length 50 with sum zero and sum of absolute values 2n.

Original entry on oeis.org

1, 4, 9, 25, 49, 121, 225, 484, 900, 1764, 3136, 5929, 10201, 18225, 30976, 53361, 88209, 148225, 240100, 393129, 627264, 1004004, 1575025, 2480625, 3833764, 5934093, 9060082, 13823450, 20838976, 31404082, 46810997, 69700899, 102868926

Offset: 1

R. H. Hardin Mar 13 2009

nonn

a(n) = A000041(n)^2 for n<=25

a(n) = A000041(n)^2 - cumulative A000712(2*n-1-length), 0 <= 2*n-1-length <= floor(n/2) [empirical].

R. H. Hardin, Table of n, a(n) for n=1..191

A229706 Triangular array read by rows: T(n,k) is the number of unimodal compositions of n with greatest part equal to k; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 5, 2, 1, 1, 9, 9, 5, 2, 1, 1, 12, 16, 10, 5, 2, 1, 1, 16, 25, 19, 10, 5, 2, 1, 1, 20, 39, 32, 20, 10, 5, 2, 1, 1, 25, 56, 54, 35, 20, 10, 5, 2, 1, 1, 30, 80, 84, 61, 36, 20, 10, 5, 2, 1, 1, 36, 109, 129, 99, 64, 36, 20, 10, 5, 2, 1

Offset: 1

Geoffrey Critzer, Sep 27 2013

A unimodal composition is a composition such that for some j, m, 1 <= x(1) <= x(2) <= ... <= x(j) >= x(j+1) >= ... >= x(m) >= 1.
Row sums are A001523.
T(2*n+1,n+1) = A000712(n) for n>=0. - Alois P. Heinz, Oct 03 2013

			1;
1,  1;
1,  2,  1;
1,  4,  2,  1;
1,  6,  5,  2,  1;
1,  9,  9,  5,  2,  1;
1, 12, 16, 10,  5,  2,  1;
1, 16, 25, 19, 10,  5,  2, 1;
1, 20, 39, 32, 20, 10,  5, 2, 1;
1, 25, 56, 54, 35, 20, 10, 5, 2, 1;
T(5,3) = 5 because we have: 3+2 = 2+3 = 3+1+1 = 1+3+1 = 1+1+3.

E. M. Wright, Stacks, Quart. J. Math. Oxford 19 (1968) 313-320, table s(r).

Alois P. Heinz, rows n = 1..141, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46

Cf. A229707.

b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(k>0, `if`(n b(n, 1, k):
seq(seq(T(n, k), k=1..n), n=1..16);  # Alois P. Heinz, Oct 03 2013

Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[Series[x^n/(1-x^n)/Product[1-x^i,{i,1,n-1}]^2,{x,0,nn}],x],{n,1,nn}]],1]]//Grid

O.g.f. for column k: x^k/prod(i=1..k-1, 1-x^i )^2.

A301970 Heinz numbers of integer partitions with more subset-products than subset-sums.

Original entry on oeis.org

165, 273, 325, 351, 495, 525, 561, 595, 675, 741, 765, 819, 825, 931, 1045, 1053, 1155, 1173, 1425, 1485, 1495, 1575, 1625, 1653, 1683, 1771, 1785, 1815, 1911, 2025, 2139, 2145, 2223, 2275, 2277, 2295, 2310, 2415, 2457, 2465, 2475, 2625, 2639, 2695, 2805, 2945

Offset: 1

Gus Wiseman, Mar 29 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A subset-sum (or subset-product) of a multiset y is any number equal to the sum (or product) of some submultiset of y.

Numbers n such that A301957(n) > A299701(n).

			Sequence of partitions begins: (532), (642), (633), (6222), (5322), (4332), (752), (743), (33222), (862), (7322), (6422), (5332), (844), (853), (62222), (5432), (972), (8332), (53222), (963), (43322), (6333).

Cf. A000712, A003963, A056239, A108917, A276024, A284640, A292886, A296150, A299701, A299702, A299729, A301854, A301855, A301856, A301957, A301979.

Select[Range[1000],With[{ptn=If[#===1,{},Join@@Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Length[Union[Times@@@Subsets[ptn]]]>Length[Union[Plus@@@Subsets[ptn]]]]&]

cp's OEIS Frontend

A307755 Exponential convolution of partition numbers (A000041) with themselves.

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A316143 Expansion of e.g.f. Product_{k>=1} 1 / (1 - (exp(x)-1)^k)^2.

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A349153 Numbers k such that the k-th composition in standard order has sum equal to twice its reverse-alternating sum.

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A355386 Position of first appearance of n in A355382, where A355382(m) = number of divisors d of m such that bigomega(d) = omega(m); or a(n) = -1 if n does not appear in A355382.

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A000715 Number of partitions of n, with three kinds of 1,2 and 3 and two kinds of 4,5,6,....

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A100535 Number of partitions of 2*n + 1 into parts of two kinds.

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A158139 Number of nondecreasing integer sequences of length 5 with sum zero and sum of absolute values 2n.

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A158184 Number of nondecreasing integer sequences of length 50 with sum zero and sum of absolute values 2n.

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