A265251 -id:A265251 - cp's OEIS frontend

A266477 Triangle read by rows in which T(n,k) is the number of partitions of n with product of multiplicities of parts equal to k; n>=0, 1<=k<=A266480(n).

1, 1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 4, 2, 2, 2, 0, 1, 5, 4, 2, 1, 1, 1, 1, 6, 6, 2, 3, 1, 2, 0, 2, 8, 7, 4, 4, 1, 2, 1, 0, 2, 1, 10, 8, 6, 6, 3, 2, 1, 3, 0, 1, 0, 2, 12, 13, 6, 6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1, 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0

Offset: 0

Emeric Deutsch and Alois P. Heinz, Dec 29 2015

Sum of entries in row n = A000041(n) = number of partitions of n.
T(n,1) = A000009(n) = number of partitions of n into distinct parts.
T(n,2) = A090858(n).
T(n,3) = A265251(n).
Smallest row m >= 0 with T(m,n) > 0 is A266325(n).
T(n,A266480(n)) gives A266871(n).

			Row 4 is [2,2,0,1]. Indeed, the products of the multiplicities of the parts in the partitions [4], [1,3], [2,2], [1,1,2], [1,1,1,1] are 1, 1, 2, 2, 4, respectively.
Triangle T(n,k) begins:
00 :  1;
01 :  1;
02 :  1,  1;
03 :  2,  0, 1;
04 :  2,  2, 0,  1;
05 :  3,  2, 1,  0, 1;
06 :  4,  2, 2,  2, 0, 1;
07 :  5,  4, 2,  1, 1, 1, 1;
08 :  6,  6, 2,  3, 1, 2, 0, 2;
09 :  8,  7, 4,  4, 1, 2, 1, 0, 2, 1;
10 : 10,  8, 6,  6, 3, 2, 1, 3, 0, 1, 0, 2;
11 : 12, 13, 6,  6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1;
12 : 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0, 2, 0, 0, 0, 2, 0, 1;

Alois P. Heinz, Rows n = 0..50, flattened

Columns k=1-10 give: A000009, A090858, A265251, A266687, A266688, A266689, A266690, A266691, A266692, A266693.
Main diagonal gives A266499.
Row lengths give A266480.
Cf. A000041, A077285, A266325, A266871.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      x^max(p, p*n), add(b(n-i*j, i-1, max(p, p*j)), j=0..n/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 1)):
seq(T(n), n=0..16);

Map[Table[Length@ Position[#, k], {k, Max@ #}] &, #] &@ Table[Map[Times @@ Map[Last, Tally@ #] &, IntegerPartitions@ n], {n, 12}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, x^Max[p, p*n], Sum[b[n - i*j, i - 1, Max[p, p*j]], {j, 0, n/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p,x]}]][ b[n, n, 1]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Sum_{k>=1} k*T(n,k) = A077285(n).
G.f. of column p if p is prime: Sum_{k>0} x^(p*k)/(1+x^k) * Product_{i>0} (1+x^i), giving the number of partitions of n such that there is exactly one part which occurs p times, while all other parts occur only once.
If p is prime then column p is asymptotic to 3^(1/4) * c(p) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)), where c(p) = Sum_{j>=0} (-1)^j/(j+p) = (PolyGamma((p+1)/2) - PolyGamma(p/2))/2. - Vaclav Kotesovec, May 24 2018

A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 2, 4, 6, 7, 8, 13, 15, 21, 25, 30, 39, 50, 58, 74, 89, 105, 129, 156, 185, 221, 264, 309, 366, 433, 505, 593, 696, 805, 941, 1090, 1258, 1458, 1684, 1933, 2225, 2555, 2922, 3346, 3823, 4349, 4961, 5644, 6402, 7267, 8234, 9309, 10525, 11886, 13393

Offset: 0

Vladeta Jovovic, Feb 12 2004

Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.

Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1], [3,3,1], [3,2,2] and [3,1,1,1,1]. - Emeric Deutsch, Apr 18 2006

			a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(2) = 1 through a(11) = 13 partitions described in the name are the following (empty columns not shown). The Heinz numbers of these partitions are given by A060687.
  (11)  (22)   (221)  (33)   (322)   (44)    (441)   (55)    (443)
        (211)  (311)  (411)  (331)   (332)   (522)   (433)   (533)
                             (511)   (422)   (711)   (442)   (551)
                             (3211)  (611)   (3321)  (622)   (722)
                                     (3221)  (4221)  (811)   (911)
                                     (4211)  (4311)  (5221)  (4322)
                                             (5211)  (5311)  (4331)
                                                     (6211)  (4421)
                                                             (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)
The a(2) = 1 through a(10) = 8 partitions described in Emeric Deutsch's comment are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325284.
  (2)  (22)  (32)   (222)   (322)    (332)     (432)      (3322)
       (31)  (311)  (3111)  (331)    (431)     (3222)     (3331)
                            (421)    (2222)    (4221)     (22222)
                            (31111)  (3311)    (4311)     (42211)
                                     (4211)    (33111)    (43111)
                                     (311111)  (42111)    (331111)
                                               (3111111)  (421111)
                                                          (31111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A047967, A265251.
Column k=2 of A266477.
Cf. A008284, A046660, A060687, A116608, A117571, A127002, A325241, A325244.

g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i,i=1..k),k=1..15): gser:=series(g,x=0,64): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 18 2006
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+3-2*t)/2, 0,
     `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
     `if`(t=1 or 2*i>n, 0, b(n-2*i, i-1, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2015

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 3 - 2*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0,  b[n - i, i - 1, t] + If[t == 1 || 2*i > n, 0, b[n - 2*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==1&]],{n,0,30}] (* Gus Wiseman, Apr 19 2019 *)

alist(n)=concat([0,0],Vec(sum(k=1,n\2,(x^(2*k)+x*O(x^n))/(1+x^k)*prod(j=1,n-2*k,1+x^j+x*O(x^n))))) \\ Franklin T. Adams-Watters, Nov 02 2015

G.f.: Sum_{k>0} x^(2*k)/(1+x^k) * Product_{k>0} (1+x^k). Convolution of 1-A048272(n) and A000009(n). a(n) = A036469(n) - A015723(n).
G.f.: sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch, Apr 18 2006
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (1 - log(2)) / (2*Pi) = 0.064273294789... - Vaclav Kotesovec, May 24 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004

a(0) added by Franklin T. Adams-Watters, Nov 02 2015

A090867 Number of partitions of n such that the set of even parts has only one element.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 9, 13, 18, 23, 32, 42, 55, 69, 89, 112, 141, 175, 217, 266, 326, 396, 480, 581, 697, 834, 996, 1183, 1402, 1660, 1954, 2297, 2694, 3150, 3674, 4280, 4970, 5762, 6669, 7701, 8876, 10219, 11737, 13460, 15418, 17628, 20125, 22951, 26128, 29709

Offset: 0

Vladeta Jovovic, Feb 12 2004

Conjecture: a(n) is also the difference between the number of parts in the odd partitions of n and the number of parts in the distinct partitions of n (offset 0). For example, if n = 5, there are 9 parts in the odd partitions of 5 (5, 311, 11111) and 5 parts in the distinct partitions of 5 (5, 41, 32), with difference 4. - George Beck, Apr 22 2017
George E. Andrews has kindly informed me that he has proved this conjecture and the result will be included in his article "Euler's Partition Identity and Two Problems of George Beck" which will appear in The Mathematics Student, 86, Nos. 1-2, January - June (2017). - George Beck, Apr 23 2017
a(n) is the number of partitions of n with exactly one repeated part. - Andrew Howroyd, Feb 14 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, Refinements of Beck-type partition identities, arXiv:2204.00105 [math.CO], 2022.
George E. Andrews, Euler's Partition Identity and Two Problems of George Beck, The Mathematics Student, 86, 1-2:115-119 (2017); Preprint.
Cristina Ballantine and Richard Bielak, Combinatorial proofs of two Euler type identities due to Andrews, arXiv:1803.06394 [math.CO], 2018.
Cristina Ballantine and Amanda Welch, Beck-type identities for Euler pairs of order r, arXiv:2006.02335 [math.NT], 2020.
Cristina Ballantine and Amanda Welch, Beck-type identities: new combinatorial proofs and a theorem for parts congruent to t mod r, arXiv:2011.08220 [math.CO], 2020.
Cristina Ballantine and Amanda Welch, Beck-type companion identities for Franklin's identity, arXiv:2101.06260 [math.CO], 2021.
Cristina Ballantine and Amanda Welch, Beck-type identities: new combinatorial proofs and a modular refinement, Ramanujan J. (2021).
Cristina Ballantine and Mircea Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts, Ramanujan J., 54:1 (2021), 107-112.
Cristina Ballantine, Hannah E. Burson, Amanda Folsom, Chi-Yun Hsu, Isabella Negrini and Boya Wen, On a Partition Identity of Lehmer, arXiv:2109.00609 [math.CO], 2021.
Cristina Ballantine and Amanda Folsom, On the number of parts in all partitions enumerated by the Rogers-Ramanujan identities, arXiv:2303.03330 [math.NT], 2023.
Shishuo Fu and Dazhao Tang, Generalizing a partition theorem of Andrews, arXiv:1705.05046 [math.CO], 2017.
Gabriel Gray, Emily Payne, and Ren Watson, Generalized partition identities and fixed perimeter analogues, Oregon State Univ. (2024). See pp. 2, 49.
Gabriel Gray, David Hovey, Brandt Kronholm, Emily Payne, Holly Swisher, and Ren Watson, A generalization of Franklin's partition identity and a Beck-type companion identity, arXiv:2410.17378 [math.NT], 2024. See p. 12. See also Ramanujan J. (2025) Volume 67, Art. No. 100.
Gabriel Gray, Emily Payne, Holly Swisher, and Ren Watson, Fixed perimeter analogues of some partition results, arXiv:2502.12394 [math.CO], 2025. See p. 15.
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018. Also Discrete Masth., 343 (2020), # 111875.
Runqiao Li and Andrew Y. Z. Wang, The dual form of Beck type identities, Ramanujan J. (2021).
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75.
Mircea Merca, On the partitions into distinct parts and odd parts, arXiv:2005.03619 [math.CO], 2020.
Aritro Pathak, On certain partition bijections related to Euler's partition problem, arXiv:2004.03596 [math.CO], 2020. Also Discrete Mathematics 345.2 (2022): 112673.
Jane Y. X. Yang, Combinatorial proofs and generalizations on conjectures related with Euler's partition theorem, arXiv:1801.06815 [math.CO], 2018.

Cf. A038348, A265251, A341494, A341495, A341496, A341497.

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
      b(n, i-1, t)+`if`(i>n or t=1 and i::even, 0,
      add(b(n-i*j, i-1, `if`(i::even, 1, t)), j=1..n/i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jun 17 2016
A090867 := proc(n)
    add(numtheory[tau](k)*A000009(n-2*k),k=1..n/2) ;
end proc: # R. J. Mathar, Jun 18 2016

f[n_] := Count[ Plus @@@ Mod[ Union /@ IntegerPartitions[n] + 1, 2], 1]; Table[ f[n], {n, 0, 50}] (* Robert G. Wilson v, Feb 16 2004 *)
a[n_] := Sum[DivisorSigma[0, k] PartitionsQ[n-2k], {k, 1, n/2}];
a /@ Range[0, 70] (* Jean-François Alcover, May 24 2021, after R. J. Mathar *)

seq(n)={Vec(sum(k=1, n\2, x^(2*k)/(1-x^(2*k)) + O(x*x^n))/prod(k=1, n\2, 1-x^(2*k-1) + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Feb 13 2021

G.f.: Sum_{m>0} x^(2*m)/(1-x^(2*m))/Product_{m>0} (1-x^(2*m-1)).
a(n) ~ 3^(1/4) * (2*gamma + log(3*n/Pi^2)) * exp(Pi*sqrt(n/3)) / (8*Pi*n^(1/4)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018
a(n) = A341494(n) + A341495(n) = A341496(n) + A341497(n). - Andrew Howroyd, Feb 14 2021

More terms from Robert G. Wilson v, Feb 16 2004

cp's OEIS Frontend

A266477 Triangle read by rows in which T(n,k) is the number of partitions of n with product of multiplicities of parts equal to k; n>=0, 1<=k<=A266480(n).

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A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

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A090867 Number of partitions of n such that the set of even parts has only one element.

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