A005895 -id:A005895 - cp's OEIS frontend

A003406 Expansion of Ramanujan's function R(x) = 1 + Sum_{n >= 1} { x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n)) }.

1, 1, -1, 2, -2, 1, 0, 1, -2, 0, 2, 0, -1, -2, 2, 1, 0, -2, 2, -2, 0, 0, 3, 0, -2, -2, 1, 0, 2, 0, 0, 0, -2, 0, 0, 1, 0, 0, 0, 2, -1, 0, -2, -2, 0, 4, 0, 2, -2, 0, -2, -1, 2, 0, -2, 2, 0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, 4, 2, -1, 0, 0, -2, -2, -2, 2, 1, 2, 0, 0, 0, 0, -2, 2, 0, 0, -2, 2, -2, -2, 0, 3, 0, 0, 2, 0, 0, 0, -2, 1, -2, 0, -2, 0

Offset: 0

N. J. A. Sloane

a(n) = A117192(n) - A117193(n) for n>0 (number of partitions into distinct parts with even rank minus those with odd rank); see also A000025. - Reinhard Zumkeller, Mar 03 2006

Ramanujan showed that R(x) = 2*Sum_{n>=0} (S(x) - P(n,x)) - 2*S(x)*D(x), where P(n,x) = Product_{k=1..n} (1+x^k), S(x) = g.f. A000009 = P(oo,x) and D(x) = -1/2 + Sum_{n>=1} x^n/(1-x^n) = -1/2 + g.f. A000005. - Michael Somos

			1 + x - x^2 + 2*x^3 - 2*x^4 + x^5 + x^7 - 2*x^8 + 2*x^10 - x^12 - 2*x^13 + ...
q + q^25 - q^49 + 2*q^73 - 2*q^97 + q^121 + q^169 - 2*q^193 + 2*q^241 - ...

G. E. Andrews, Ramanujan's "lost" notebook V: Euler's partition identity, Adv. in Math. 61 (1986), no. 2, 156-164; Math. Rev. 87i:11137. [ The expansion in (2.8) is incorrect. ]
F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
F. J. Dyson, Selected Papers, Am. Math. Soc., 1996, p. 200.
B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989. see page 182. MR1019331 (90k:11050)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2000 from T. D. Noe)
G. E. Andrews, Questions and conjectures in partition theory, Amer. Math. Monthly, 93 (1986), 708-711.
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000.
G. E. Andrews, F. J. Dyson and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988) 391-407.
S.-Y. Kang, Generalizations of Ramanujan's reciprocity theorem and their applications, J. London Math. Soc., 75 (2007), 18-34.
Alexander E. Patkowski, A note on the rank parity function, Discrete Math. 310 (2010), 961-965.
D. Zagier, Quantum modular forms, Example 1 in Quanta of Maths: Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675

Cf. A003475, A005895, A005896, A158690.

g:=1+sum(x^(n*(n+1)/2)/product(1+x^j,j=1..n),n=1..20): gser:=series(g,x=0,110): seq(coeff(gser,x,n),n=0..104); # Emeric Deutsch, Mar 30 2006
t1:= add( (-1)^n*q^(n*(3*n+1)/2)*(1-q^(2*n+1))* add( (-1)^j*q^(-j^2),j=-n..n), n=0..20); t2:=series(t1,q,40); # N. J. A. Sloane, Jun 27 2011

max = 105; f[x_] := 1 + Sum[ x^(n*(n+1)/2) / Product[ 1+x^j, {j, 1, n}], {n, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 02 2011 *)
max = 105; s = 1 + Sum[2*q^(n*(n+1)/2)/QPochhammer[-1, q, n+1], {n, 1, Ceiling[Sqrt[2 max]]}] + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

{a(n) = local(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum( k=1, n, t *= if( k>1, x^k - x, x) + O(x^(n-k+2)), 1), n))} /* Michael Somos, Mar 07 2006 */

{a(n) = local(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum( k=1, (sqrtint(8*n + 1)-1)\2, t *= x^k / (1 + x^k) + x * O(x^(n - (k^2-k)/2)), 1), n))} /* Michael Somos, Aug 17 2006 */

{a(n) = local(A, p, e, x, y); if( n<0, 0, n = 24*n+1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p<5, 0, if( p%24>1 && p%24<23, if(e%2, 0, if( p%24==7 || p%24==17, (-1)^(e/2), 1)), x=y=0; if( p%24==1, forstep(i=1, sqrtint(p), 2, if( issquare( (i^2+p)/2, &y), x=i; break)), for( i=1, sqrtint(p\2), if( issquare(2*i^2 + p, &x), y=i; break))); (e+1)*(-1)^( (x + if((x-y)%6, y, -y))/6*e))))))} /* Michael Somos, Aug 17 2006 */

G.f.: 1 - Sum_{n > 0} (-x)^n * (1 - x) * (1 - x^2) * ... * (1  -x^(n-1)).
G.f.: 1 + Sum_{n>=1}(x^(n(n+1)/2)/Product_{j=1..n}(1+x^j)). - Emeric Deutsch, Mar 30 2006
Define c(24*k + 1) = A003406(k), c(24*k - 1) = -2*A003475(k), c(n) = 0 otherwise. Then c(n) is multiplicative with c(2^e) = c(3^e) = 0^e, c(p^e) = (-1)^(e/2) * (1+(-1)^e)/2 if p == 7, 17 (mod 24), c(p^e) = (1+(-1)^e)/2 if p == 5, 11, 13, 19 (mod 24), c(p^e) = (e+1)*(-1)^(y*e) where p == 1, 23 (mod 24) and p = x^2 - 72*y^2 . - Michael Somos, Aug 17 2006
Also R(x) = -2 + Sum_{n>=0} (n+1)*x^(n(n-1)/2)/(Product_{k=1..n} (1+x^k)). - Paul D. Hanna, May 22 2010

A092265 Sum of smallest parts of all partitions of n into distinct parts.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 14, 16, 23, 26, 34, 40, 50, 58, 74, 83, 102, 120, 142, 164, 198, 226, 266, 308, 359, 412, 482, 548, 634, 730, 834, 950, 1094, 1240, 1416, 1609, 1826, 2068, 2350, 2648, 2994, 3382, 3806, 4280, 4826, 5408, 6070, 6806, 7619, 8522, 9534, 10632

Offset: 1

Vladeta Jovovic, Feb 14 2004

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A046746, A005895, A006128, A092269, A092319, A092316, A034296, A237665.

Cf. A026832, A336902, A336903.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n,i+1)+b(n-i, i+1)))
    end:
a:= n-> add(j*b(n-j, j+1), j=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i + 1] + b[n - i, i + 1]]]; a[n_] := Sum[j*b[n - j, j + 1], {j, 1, n}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 21 2017, after Alois P. Heinz *)

G.f.: Sum_{n >= 1} (-1 + Product_{k >= n} 1 + x^k).
G.f.: Sum_{n >= 1} n*x^n*Product_{k >= n+1} (1 + x^k). - Joerg Arndt, Jan 29 2011
G.f.: Sum_{k >= 1} x^(k*(k+1)/2)/(1 - x^k)/Product_{i = 1..k} (1 - x^i). - Vladeta Jovovic, Aug 10 2004
Conjecture: a(n) = A034296(n) + A237665(n+1). - George Beck, May 06 2017
a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 20 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A336902 Sum of the smallest parts of all compositions of n into distinct parts.

Original entry on oeis.org

0, 1, 2, 5, 6, 11, 18, 25, 32, 53, 84, 107, 156, 205, 302, 497, 618, 863, 1206, 1597, 2228, 3569, 4440, 6191, 8256, 11329, 14642, 20477, 30390, 38555, 52578, 69625, 92696, 122141, 160500, 211955, 310476, 386941, 521102, 678617, 901386, 1155383, 1529742, 1940749

Offset: 0

Alois P. Heinz, Aug 07 2020

nonn

			a(6) = 18 = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 1 + 1 + 6: (1)23, (1)32, 2(1)3, 23(1), 3(1)2, 32(1), (2)4, 4(2), (1)5, 5(1), (6).

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A005895, A006128, A046746, A092265, A097939, A102712, A336903.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..50);

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n || i < 1, 0,
     If[i == n, i*p!, b[n-i, Min[n-i, i-1], p+1]] + b[n, i-1, p]];
a[n_] := b[n, n, 1];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)

a(n) == n (mod 2).

A336903 Sum of the largest parts of all compositions of n into distinct parts.

Original entry on oeis.org

0, 1, 2, 7, 10, 19, 42, 61, 98, 151, 304, 403, 654, 925, 1400, 2431, 3328, 4903, 7056, 10117, 13952, 23419, 30406, 44683, 61308, 87289, 116822, 164359, 247774, 327715, 457542, 624445, 855062, 1148023, 1559188, 2058643, 3043506, 3906637, 5375732, 7111975, 9679852

Offset: 0

Alois P. Heinz, Aug 07 2020

nonn

			a(6) = 42 = 3 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 5 + 5 + 6: 12(3), 1(3)2, 21(3), 2(3)1, (3)12, (3)21, 2(4), (4)2, 1(5), (5)1, (6).

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A005895, A006128, A046746, A092265, A097939, A102712, A336902.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, b(n$2, 0)):
seq(a(n), n=0..50);

b[n_, i_, p_] := b[n, i, p] = If[i(i + 1)/2 < n, 0,
     If[n == 0, p!, b[n - i, Min[n - i, i - 1], p + 1]*
     If[p == 0, i, 1] + b[n, i - 1, p]]];
a[n_] := If[n == 0, 0, b[n, n, 0]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)

a(n) == n (mod 2).

A092316 Sum of largest parts of all partitions of n into odd distinct parts.

Original entry on oeis.org

1, 0, 3, 3, 5, 5, 7, 12, 14, 16, 18, 27, 29, 33, 42, 55, 59, 65, 78, 95, 110, 118, 137, 167, 188, 200, 236, 274, 303, 330, 376, 435, 485, 522, 591, 677, 741, 803, 903, 1022, 1115, 1210, 1345, 1505, 1650, 1784, 1964, 2201, 2393, 2578, 2843, 3143, 3409, 3685, 4034

Offset: 1

Vladeta Jovovic, Feb 15 2004

			a(13) = 29 because the partitions of 13 into distinct odd parts are [13],[9,3,1] and [7,5,1], with sum of largest terms 13+9+7 = 29.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Seiichi Manyama)
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Cf. A000700, A005895, A067619, A092319, A116857.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1 or i^2 `if`(t>n, 0, b(n-t, i-1)))(2*i-1) ))
    end:
a:= n-> add(`if`(j::odd, j*b(n-j, (j-1)/2), 0), j=1..n):
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 19 2022

nmax = 50; Rest[CoefficientList[Series[Sum[(2*k - 1)*x^(2*k - 1) * Product[1 + x^(2*j - 1), {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 28 2016 *)

G.f.: Sum_{n>=1} (2*n-1)*x^(2*n-1)*Product_{k=1..n-1} (1+x^(2*k-1)).

a(n) = 2 * A067619(n) - A000700(n). - Seiichi Manyama, Jan 19 2022

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A005896 Weighted count of partitions with odd parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 19, 26, 34, 45, 59, 76, 96, 121, 153, 189, 234, 288, 353, 428, 519, 625, 752, 900, 1073, 1274, 1512, 1784, 2101, 2470, 2894, 3382, 3946, 4590, 5330, 6179, 7144, 8246, 9505, 10931, 12552, 14396, 16476, 18831, 21495

Offset: 0

N. J. A. Sloane, Simon Plouffe

			G.f. = x^3 + x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 9*x^8 + 14*x^9 + 19*x^10 + ... - _Michael Somos_, Oct 21 2018

Andrews, George E. Ramanujan's "lost" notebook. V. Euler's partition identity. Adv. in Math. 61 (1986), no. 2, 156-164.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000009, A005895, A003406.

max = 48; f[n_, x_] := Product[ 1/(1-x^(2k+1)), {k, 0, n}]; g[x_] = Sum[ f[max/2, x] - f[n, x], {n, 0, max/2}]; CoefficientList[ Series[ g[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 17 2011, after g.f. *)
a[ n_] := With[{A = 1 / QPochhammer[ q, q^2]}, SeriesCoefficient[ Sum[A - 1 / QPochhammer[ q, q^2, k], {k, 1, n/2}], {q, 0, n}]]; (* Michael Somos, Oct 21 2018 *)

/* set maximum */ MM = 50; /* G.f. for partitions with odd parts: */ (Q(n, q) = prod(k=0, n, 1/(1 - q^(2*k+1)), 1 + q*O(q^MM))); /* G.f. for A000009: */ Sq = Q(MM/2, q); /* G.f. for A005896: */ Sq0 = sum(n=0, MM/2, Sq-Q(n, q)); for(n=0, 48, print1(polcoeff(Sq0, n)","));

G.f.: Sum_{n=0..infinity} {S(q)-1/((1-q)(1-q^3)...(1-q^(2n+1)))}, where S(q) = g.f. for A000009.

More terms from Michael Somos.

A117455 Sum of the differences between the largest part and smallest part over all partitions of n into distinct parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 12, 19, 27, 41, 54, 76, 99, 133, 171, 223, 279, 357, 443, 554, 682, 841, 1022, 1247, 1504, 1814, 2174, 2603, 3092, 3676, 4346, 5127, 6030, 7076, 8275, 9669, 11254, 13078, 15167, 17556, 20270, 23377, 26899, 30902, 35448, 40592, 46403

Offset: 1

Emeric Deutsch, Mar 18 2006

nonn

a(n) = sum(k*A117454(n,k), k=0..n-2).

a(n) = A005895(n)-A092265(n). - Alois P. Heinz, Jul 06 2012

			a(7)=12 because the partitions of 7 into distinct parts are [7], [6,1], [5,2], [4,3] and [4,2,1] and (7-7)+(6-1)+(5-2)+(4-3)+(4-1)=12.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A005895, A092265, A117454.

g:=sum(x^(i*(i+1)/2)*sum(1/(1-x^j),j=1..i-1)/product(1-x^j,j=1..i),i=1..15): gser:=series(g,x=0,55): seq(coeff(gser,x^n), n=1..50);
# second Maple program:
b:= proc(n, i) option remember;
      `if`(i=n, n, 0)+`if`(i>0, b(n, i-1)+
      `if`(i g(n, 1) -b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 06 2012

b[n_, i_] := b[n, i] = If[i==n, n, 0] + If[i>0, b[n, i-1] + If[iJean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

G.f.: sum(x^(i(i+1)/2)*sum(1/(1-x^j), j=1..i-1)/product(1-x^j, j=1..i), i=1..infinity) (obtained by taking the derivative with respect to t of the g.f. G(t,x) of A117454 and letting t=1).

cp's OEIS Frontend

A003406 Expansion of Ramanujan's function R(x) = 1 + Sum_{n >= 1} { x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n)) }.

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A092265 Sum of smallest parts of all partitions of n into distinct parts.

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A336902 Sum of the smallest parts of all compositions of n into distinct parts.

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A336903 Sum of the largest parts of all compositions of n into distinct parts.

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A092316 Sum of largest parts of all partitions of n into odd distinct parts.

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A005896 Weighted count of partitions with odd parts.

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A117455 Sum of the differences between the largest part and smallest part over all partitions of n into distinct parts.

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