A006906 -id:A006906 - cp's OEIS frontend

A118851 Product of parts in n-th partition in Abramowitz and Stegun order.

1, 1, 2, 1, 3, 2, 1, 4, 3, 4, 2, 1, 5, 4, 6, 3, 4, 2, 1, 6, 5, 8, 9, 4, 6, 8, 3, 4, 2, 1, 7, 6, 10, 12, 5, 8, 9, 12, 4, 6, 8, 3, 4, 2, 1, 8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1, 9, 8, 14, 18, 20, 7, 12, 15, 16, 20, 24, 27, 6, 10, 12, 16, 18, 24, 5, 8, 9, 12, 16, 4

Offset: 0

Alford Arnold, May 01 2006

Let Theta(n) denote the set of norm values corresponding to all the partitions of n. The following results hold regarding this set: (i) Theta(n) is a subset of Theta(n+1); (ii) A prime p will appear as a norm only for partitions of n>=p; (iii) There exists a prime p not in Theta(n) for all n>=6; (iv) Let h(k) be the prime floor function which gives the greatest prime less than or equal to the k, then the prime p=h(n+1) does not belong to Theta(n); and (v) The primes not in the set Theta(n) are A000720(A000792(n)) - A000720(n). - Abhimanyu Kumar, Nov 25 2020

			a(9) = 4 because the 9th partition is [2,2] and 2*2 = 4.
Table T(n,k) starts:
  1;
  1;
  2, 1;
  3, 2,  1;
  4, 3,  4,  2,  1;
  5, 4,  6,  3,  4, 2,  1;
  6, 5,  8,  9,  4, 6,  8,  3,  4,  2, 1;
  7, 6, 10, 12,  5, 8,  9, 12,  4,  6, 8, 3, 4,  2,  1;
  8, 7, 12, 15, 16, 6, 10, 12, 16, 18, 5, 8, 9, 12, 16, 4, 6, 8, 3, 4, 2, 1;

Abramowitz and Stegun, Handbook (1964) page 831.

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Walter Bridges and William Craig, On the distribution of the norm of partitions, arXiv:2308.00123 [math.CO], 2023.
Abhimanyu Kumar and Meenakshi Rana, On the treatment of partitions as factorization and further analysis, Journal of the Ramanujan Mathematical Society 35(3), 263-276 (2020).
Wolfdieter Lang, Rows n=1..10.
Robert Schneider and Andrew V. Sills, The Product of Parts or "Norm" of a Partition, INTEGERS, Volume 20A (2020) Paper #A13, 16 pp.
Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019-2021.

Cf. A000041 (row lengths), A006906 (row sums).

Cf. A005361, A036035, A036036, A048996, A078812, A085643, A103921, A111786.

C(sig)={vecprod(sig)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 19 2020

a(n) = A085643(n)/A048996(n).

T(n,k) = A005361(A036035(n,k)). - Andrew Howroyd, Oct 19 2020

Corrected and extended by Franklin T. Adams-Watters, May 26 2006

A162506 Convergent of an infinite product, abc,...; a = [1,1,1,...], b = [1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...],...

Original entry on oeis.org

1, 1, 3, 6, 12, 23, 42, 77, 132, 236, 390, 664, 1087, 1782, 2858, 4601, 7216, 11344, 17650, 27162, 41632, 63316, 95717, 143558, 214644, 318464, 470879, 691968, 1012866, 1474434, 2140606, 3088874, 4445440, 6370142, 9095564, 12941289, 18350398, 25930984

Offset: 1

Gary W. Adamson, Jul 04 2009

nonn

Equals row sums of triangle A162507.

With offset 0, sum of products of parts, counted without multiplicity, in all partitions of n. Sum of products of parts, counted with multiplicity, in all partitions of n is A006906. - Vladeta Jovovic, Jul 24 2009

			First few rows of the array =
1,...1,...1,...1,...1,...
1,...1,...3,...3,...5,...
1,...1,...3,...6,...8,...
1,...1,...3,...6,..12,...
1,...1,...3,...6,..12,...
...tending to A162506: (1, 1, 3, 6, 12, 23, 42, 77, 132,...)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A162507, A267007.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-1)*i, j=1..n/i)))
    end:
a:= n-> b(n-1, n-1):
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 26 2013

nmax = 50; Rest[CoefficientList[Series[x*Product[1+k*x^k/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 08 2016 *)

Convergent of an infinite product, a*b*c,...; a = [1,1,1,...], b =
[1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...]; i.e. the infinite set of
sequences [1,...N,...,] interleaved with (N-2) adjacent zeros.
G.f.: x*Product(1+k*x^k/(1-x^k),k=1..infinity). - Vladeta Jovovic, Jul 24 2009

More terms from Vladeta Jovovic, Jul 22 2009

A265837 Expansion of Product_{k>=1} 1/(1 - k^3*x^k).

Original entry on oeis.org

1, 1, 9, 36, 164, 505, 2474, 7273, 31008, 103644, 379890, 1226802, 4747529, 14553648, 52167558, 171639695, 583371802, 1851395692, 6427705062, 19983302144, 67235043192, 214615427776, 697704303005, 2194982897304, 7262755260410, 22402942281766, 72461661415093

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A006906, A077335, A265838, A265839, A265840.

Column k=3 of A292193.

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

a(n) ~ c * 3^n, where
c = 86.60286320343345379122228784466307940393110978... if n mod 3 = 0
c = 86.27536745612304663727011387030370600864018892... if n mod 3 = 1
c = 86.29819842537784019895326532818285333403267092... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(3*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

A285241 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^2).

Original entry on oeis.org

1, 1, 9, 36, 140, 481, 1774, 5925, 20076, 64980, 208486, 652058, 2017023, 6117878, 18347256, 54222195, 158463794, 457570786, 1307951914, 3700153918, 10371860026, 28810051738, 79359812567, 216834266612, 587961817595, 1582612248239, 4230325722508

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

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

Cf. A006906, A023871, A266941, A285240, A285243.

nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^(k^2), {k,1,nmax}], {x,0,nmax}], x]

a(n) ~ c * n^8 * 3^(n/3), where
if mod(n,3) = 0 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560142\
40331306860864399770618296475558098172993864629247911801570500913143642\
65158886200452165335605783203726486071335...
if mod(n,3) = 1 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560112\
77299895134841028015999951571187798033179513268954711586617617334007687\
07198348808962592621276659532114355538024...
if mod(n,3) = 2 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560117\
00278534968233203470801053870003971422069097966617636511346003845666735\
79293861331368526745743422198017148868212...
In closed form, a(n) ~ -(27*Product_{k>=4}((1 - k / 3^(k/3))^(-k^2)) / (13 + 128*3^(1/3) - 95*3^(2/3)) + 243*Product_{k>=4}((1 + (-1)^(1 + 2*k/3) * k / 3^(k/3))^(-k^2)) / ((-1)^(2*n/3) * ((3 + 2*(-3)^(1/3))^4 * (-3 + (-3)^(2/3)))) + (-1)^(1 - 4*n/3) * Product_{k>=4}((1 + (-1)^(1 + 4*k/3) * k / 3^(k/3))^(-k^2)) / ((1 + (-1/3)^(1/3)) * (1 - 2*(-1/3)^(2/3))^4)) / 793618560 * n^8 * 3^(n/3).

A300277 G.f.: 1 + Sum_{n>=1} a(n)x^n/(1 - x^n) = Product_{n>=1} 1/(1 - nx^n).

Original entry on oeis.org

1, 2, 5, 11, 24, 48, 96, 184, 348, 645, 1169, 2140, 3761, 6687, 11645, 20326, 34635, 59854, 100579, 171211, 285718, 479325, 791315, 1318955, 2156805, 3553589, 5783306, 9445861, 15250215, 24759156, 39713787, 63991400, 102197851, 163548416, 259744930, 413761633, 653715967

Offset: 1

Ilya Gutkovskiy, Mar 01 2018

nonn

Moebius transform of A006906.

Vaclav Kotesovec, Table of n, a(n) for n = 1..6000
N. J. A. Sloane, Transforms

Cf. A000837, A006906, A078374, A085410, A300274, A300275, A300276, A300278.

nn = 37; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[1/(1 - n x^n), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[1/(1 - k x^k), {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 37}]

a(n) = Sum_{d|n} mu(n/d)*A006906(d).

A306901 Sum over all partitions of n of the bitwise AND of the parts.

Original entry on oeis.org

0, 1, 3, 4, 8, 9, 14, 13, 24, 28, 36, 38, 55, 54, 68, 75, 106, 120, 154, 168, 208, 228, 269, 298, 374, 404, 475, 530, 618, 682, 808, 896, 1080, 1220, 1410, 1581, 1828, 2022, 2322, 2598, 2963, 3278, 3732, 4128, 4684, 5218, 5888, 6550, 7418, 8192, 9198, 10187

Offset: 0

Alois P. Heinz, Mar 15 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bitwise operation
Wikipedia, Commutative property
Wikipedia, Identity element
Wikipedia, Partition (number theory)
Wikipedia, Truth table

Cf. A006906, A066186, A306884, A306895, A306902, A306903, A306923.

b:= proc(n, i, r) option remember; `if`(i<1, 0, (t->
      `if`(i b(n$2, 2^ilog2(2*n)-1):
seq(a(n), n=0..55);

A306902 Sum over all partitions of n of the bitwise OR of the parts.

Original entry on oeis.org

0, 1, 3, 7, 13, 23, 40, 67, 103, 156, 231, 340, 486, 689, 964, 1352, 1845, 2507, 3363, 4500, 5937, 7814, 10174, 13247, 17064, 21930, 27957, 35616, 45009, 56805, 71252, 89320, 111282, 138479, 171421, 212021, 260974, 320837, 392753, 480395, 585239, 712163, 863536

Offset: 0

Alois P. Heinz, Mar 15 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bitwise operation
Wikipedia, Commutative property
Wikipedia, Identity element
Wikipedia, Partition (number theory)
Wikipedia, Truth table

Cf. A006906, A066186, A306884, A306895, A306901, A306903, A306924.

b:= proc(n, i, r) option remember; `if`(i<1, 0, (t->
      `if`(i b(n$2, 0):
seq(a(n), n=0..45);

A306903 Sum over all partitions of n of the bitwise XOR of the parts.

Original entry on oeis.org

0, 1, 2, 7, 8, 19, 26, 61, 70, 126, 146, 270, 308, 519, 604, 1054, 1222, 1929, 2208, 3454, 3930, 5862, 6576, 9833, 11102, 16052, 17904, 25752, 28764, 40479, 44830, 62988, 70188, 97151, 107662, 148141, 164710, 223783, 247380, 334035, 370406, 495313, 547000

Offset: 0

Alois P. Heinz, Mar 15 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bitwise operation
Wikipedia, Commutative property
Wikipedia, Identity element
Wikipedia, Partition (number theory)
Wikipedia, Truth table

Cf. A006906, A066186, A067567, A306884, A306895, A306901, A306902, A306925.

b:= proc(n, i, r) option remember; `if`(i<1, 0, (t->
      `if`(i b(n$2, 0):
seq(a(n), n=0..45);

a(n) is odd <=> n in { A067567 }.

A318415 Expansion of Product_{i>=1, j>=1} 1/(1 - ijx^(i*j)).

Original entry on oeis.org

1, 1, 5, 11, 35, 69, 200, 398, 1014, 2069, 4820, 9716, 21787, 43209, 92530, 182773, 378676, 737526, 1492451, 2872788, 5686194, 10837935, 21052463, 39699970, 75972300, 141818166, 267607065, 495142606, 922920753, 1692529453, 3121105278, 5676677651, 10364752129, 18708292447, 33851433117, 60656841965

Offset: 0

Ilya Gutkovskiy, Aug 26 2018

nonn

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

Cf. A000005, A006171, A006906, A280540, A318416.

nmax = 35; CoefficientList[Series[Product[Product[1/(1 - i j x^(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Product[1/(1 - k x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Exp[Sum[Sum[d^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 35}]
nmax = 40; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*(-1)^j*k^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] (* Vaclav Kotesovec, Aug 27 2018 *)

G.f.: Product_{k>=1} 1/(1 - k*x^k)^tau(k), where tau = number of divisors (A000005).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d^(k/d+1)*tau(d) ) * x^k/k).
From Vaclav Kotesovec, Aug 27 2018: (Start)
a(n) ~ c * n * 3^(n/3), where
c = 10751825728554.298582954430359167227238488440778317... if mod(n,3)=0
c = 10751825728553.835664124121831524829543267756895348... if mod(n,3)=1
c = 10751825728553.838520991588115910603754564083195806... if mod(n,3)=2
In closed form, c = (Product_{k>=4}((1 - k/3^(k/3))^(-sigma(0,k)))) / (21 - 16*3^(1/3) + 3^(2/3)) - (3*Product_{k>=4}((1 + ((-1)^(1 + 2*k/3)*k)/3^(k/3))^(-sigma(0,k)))) / ((-1)^(2*n/3)*((3 + 2*(-3)^(1/3))^2*(-3 + (-3)^(2/3)))) + Product_{k>=4}((1 + ((-1)^(1 + 4*k/3)*k)/3^(k/3))^(-sigma(0,k))) / (9*(-1)^(4*n/3)*((1 + (-1/3)^(1/3))*(1 - 2*(-1/3)^(2/3))^2))
(End)

A336975 Expansion of Product_{k>=1} 1/(1 - x^k * (k + x)).

Original entry on oeis.org

1, 1, 4, 9, 22, 47, 107, 221, 468, 953, 1932, 3814, 7560, 14625, 28192, 53757, 101827, 190907, 356362, 659716, 1215314, 2224968, 4053914, 7346367, 13260001, 23822114, 42629786, 75991017, 134991954, 238948942, 421656911, 741750026, 1301116634, 2275985891, 3971022904

Offset: 0

Seiichi Manyama, Aug 09 2020

nonn

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

Cf. A006906, A227681, A336976, A336977, A336978, A336979, A336980.

m = 34; CoefficientList[Series[Product[1/(1 - x^k*(k + x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)

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

N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (k/d+x)^d/d))))

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (k/d + x)^d / d).

a(n) ~ c * n * phi^(n+1) / 5, where c = Product_{k>=3} 1/(1 - 1/phi^k*(k + 1/phi)) = 167.5661037860673786430316975350024960626825333609486463342... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 06 2021

cp's OEIS Frontend

A118851 Product of parts in n-th partition in Abramowitz and Stegun order.

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A162506 Convergent of an infinite product, a*b*c,...; a = [1,1,1,...], b = [1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...],...

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A265837 Expansion of Product_{k>=1} 1/(1 - k^3*x^k).

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A285241 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^2).

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A300277 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} 1/(1 - n*x^n).

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A306901 Sum over all partitions of n of the bitwise AND of the parts.

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A306902 Sum over all partitions of n of the bitwise OR of the parts.

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Maple

A306903 Sum over all partitions of n of the bitwise XOR of the parts.

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A162506 Convergent of an infinite product, abc,...; a = [1,1,1,...], b = [1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...],...

A300277 G.f.: 1 + Sum_{n>=1} a(n)x^n/(1 - x^n) = Product_{n>=1} 1/(1 - nx^n).

A318415 Expansion of Product_{i>=1, j>=1} 1/(1 - ijx^(i*j)).