A053263 -id:A053263 - cp's OEIS frontend

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A238478 Number of partitions of n whose median is a part.

1, 2, 2, 4, 5, 8, 11, 17, 22, 32, 43, 59, 78, 105, 136, 181, 233, 302, 386, 496, 626, 796, 999, 1255, 1564, 1951, 2412, 2988, 3674, 4516, 5524, 6753, 8211, 9984, 12086, 14617, 17617, 21211, 25450, 30514, 36475, 43550, 51869, 61707, 73230, 86821, 102706

Offset: 1

Clark Kimberling, Feb 27 2014

Also the number of integer partitions of n with a unique middle part. This means that either the length is odd or the two middle parts are equal. For example, the partition (4,3,2,1) has middle parts {2,3} so is not counted under a(10), but (3,2,2,1) has middle parts {2,2} so is counted under a(8). - Gus Wiseman, May 13 2023

			a(6) counts these partitions:  6, 411, 33, 321, 3111, 222, 21111, 111111.

For mean instead of median we have A237984, ranks A327473.
The complement is counted by A238479, ranks A362617.
These partitions have ranks A362618.
A000041 counts integer partitions.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
A359908 ranks partitions with integer median, complement A359912.
Cf. A002865, A008284, A053263, A238480, A240219, A327472, A359907, A360686, A360687, A362608.

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Median[p]]], {n, 40}]

a(n) + A238479(n) = A000041(n).
For all n, a(n) >= A027193(n) (because when a partition of n has an odd number of parts, its median is simply the part at the middle). - Antti Karttunen, Feb 27 2014
a(n) = A078408(n-1) - A282893(n). - Mathew Englander, May 24 2023

A053266 Coefficients of the '5th-order' mock theta function Phi(q).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 9, 10, 12, 12, 14, 15, 17, 18, 20, 21, 25, 26, 29, 31, 35, 36, 41, 43, 48, 51, 56, 59, 66, 70, 76, 81, 89, 94, 103, 109, 119, 126, 137, 144, 158, 167, 180, 191, 207, 218, 236, 250, 269, 285, 306, 323, 349, 368

Offset: 0

Dean Hickerson, Dec 19 1999

In Ramanujan's lost notebook the generating function is denoted by phi(q) on pages 18 and 20, however on page 18 there is no minus one first term. - Michael Somos, Jul 07 2015

			G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, MR2952081, See p. 12, Equation (2.1.18) and also page 26 equation (2.4.8).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20, 23.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660.

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053267.

Cf. A259910.

a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24 n/5]}, SeriesCoefficient[ -1 + Sum[ (-1)^k x^(5 k (3 k + 1)/2) / (1 - x^(5 k + 1)), {k, Quotient[m + 1, -6], Quotient[m - 1, 6]}] / QPochhammer[ x^5], {x, 0, n}]]]; (* Michael Somos, Jul 07 2015 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n\5), x^(5*k^2) / prod(i=1, 5*k+1, 1 - if( i%5==1 || i%5==4, x^i), 1 + x * O(x^(n - 5*k^2)))) - 1, n))}; /* Michael Somos, Jul 07 2015 */

{a(n) = my(A, m); if( n<0, 0, m = sqrtint(1 + 24*n\5); A = x * O(x^n); polcoeff( sum(k=(m + 1)\-6, (m - 1)\6, (-1)^k * x^(5*k*(3*k + 1)/2) / (1 - x^(5*k + 1)), A) / eta(x^5 + A) - 1, n))}; /* Michael Somos, Jul 07 2015 */

G.f.: -1 + Sum_{k>=0} q^(5k^2)/((1-q)(1-q^4)(1-q^6)(1-q^9)...(1-q^(5k+1))).
3*a(n) = A053262(n) + A259910(n) unless n=0. - Michael Somos, Jul 07 2015
a(n) ~ sqrt(phi/2) * exp(Pi*sqrt(2*n/15)) / (5^(3/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A237821 Number of partitions of n such that 2*(least part) <= greatest part.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 11, 16, 25, 35, 48, 68, 92, 123, 164, 216, 282, 367, 471, 604, 769, 975, 1225, 1542, 1924, 2395, 2968, 3669, 4514, 5547, 6781, 8280, 10071, 12229, 14796, 17881, 21537, 25902, 31066, 37206, 44443, 53021, 63098, 74995, 88946, 105350, 124533

Offset: 1

Clark Kimberling, Feb 16 2014

By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - Gus Wiseman, May 15 2023

			a(6) = 7 counts these partitions:  51, 42, 411, 321, 3111, 2211, 21111.
From _Gus Wiseman_, May 15 2023: (Start)
The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:
  (21)  (31)   (41)    (42)     (52)
        (211)  (221)   (51)     (61)
               (311)   (321)    (331)
               (2111)  (411)    (421)
                       (2211)   (511)
                       (3111)   (2221)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
The a(3) = 1 through a(8) = 16 partitions with different median from maximum:
  (21)  (31)   (32)    (42)     (43)
        (211)  (41)    (51)     (52)
               (311)   (321)    (61)
               (2111)  (411)    (322)
                       (2211)   (421)
                       (3111)   (511)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
(End)

The complement is counted by A053263, ranks A081306.
These partitions have ranks A069900.
The case of equality is A118096.
For < instead of <= we have A237820, ranks A362982.
For >= instead of <= we have A237824, ranks A362981.
The conjugate partitions have ranks A362980.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865, A008284, A171979, A237984, A238478, A238479, A327472, A359893, A362612, A362622.

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)

G.f.: Sum_{i>=1} Sum_{j>=0} x^(3*i+j) /Product_{k=i..2*i+j} (1-x^k). - Seiichi Manyama, May 27 2023

A053256 Coefficients of the '5th-order' mock theta function f_0(q).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, -1, 1, 0, 1, -2, 1, -1, 2, -2, 2, -1, 1, -3, 2, -1, 3, -3, 2, -2, 3, -4, 3, -3, 4, -5, 5, -3, 5, -7, 5, -5, 6, -7, 7, -6, 7, -9, 9, -7, 9, -11, 9, -9, 11, -13, 12, -11, 13, -15, 15, -13, 16, -19, 17, -17, 19, -21, 21, -20, 22, -26, 25, -23, 27, -30, 29, -28, 32, -35, 34, -34, 36, -41, 40, -38, 44, -48, 46

Offset: 0

Dean Hickerson, Dec 19 1999

In Ramanujan's lost notebook page 21 is written the g.f. neatly crossed out between the 3rd and 4th equations. - Michael Somos, Feb 13 2017

			G.f. = 1 + x - x^2 + x^3 - x^6 + x^7 + x^9 - 2*x^10 + x^11 - x^12 + 2*x^13 - ...

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22, 23.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Other '5th-order' mock theta functions are at A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

N:= 100: # for a(0)..a(N)
g:= add(q^(k^2)/mul(1+q^i,i=1..k),k=0..floor(sqrt(N))):
S:= series(g,q,N+1):
seq(coeff(S,q,k),k=0..N)]; # Robert Israel, Mar 27 2018

Series[Sum[q^n^2/Product[1+q^k, {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
a[ n_] := SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ -x, x, k] // FunctionExpand, {k, 0, Sqrt@ n}], {x, 0, n}]; (* Michael Somos, Feb 13 2017 *)

{a(n) = my(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum( k=1, sqrtint(n), t *= x^(2*k-1) / (1 + x^k + O(x^(n - (k-1)^2 + 1))), 1), n))}; /* Michael Somos, Mar 12 2006 */

G.f.: 1 + Sum_{k>0} q^k^2 / ((1 + q) * (1 + q^2) * ... * (1 + q^k)).
Consider partitions of n into parts differing by at least 2. For n > 0: a(n) is the number of them with largest part odd minus number with largest part even.
a(n) ~ -(-1)^n * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 15 2019

A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 19, 21, 30, 31, 42, 50, 62, 69, 91, 99, 126, 144, 175, 198, 246, 275, 331, 379, 452, 509, 612, 686, 811, 922, 1076, 1219, 1428, 1604, 1863, 2108, 2434, 2739, 3162, 3551, 4075, 4593, 5240, 5885, 6721, 7527, 8556, 9597, 10870

Offset: 0

Reinhard Zumkeller, Jan 20 2010

nonn

A000009(n) <= a(n) <= A000041(n).
Equivalently, the number of partitions of n such that (maximal multiplicity of parts) = (multiplicity of the maximal part), as in the Mathematica program. - Clark Kimberling, Apr 04 2014
Also the number of integer partitions of n whose greatest part is a mode, meaning it appears at least as many times as each of the others. The name "Number of partitions of n such that smaller parts do not occur more frequently than greater parts" seems to describe A100882 = "Number of partitions of n in which the sequence of frequencies of the summands is nonincreasing," which first differs from this at n = 10 due to the partition (3,3,2,1,1). - Gus Wiseman, May 07 2023

			a(5) = #{5, 4+1, 3+2, 2+2+1, 5x1} = 5;
a(6) = #{6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 6x1} = 8;
a(7) = #{7, 6+1, 5+2, 4+3, 4+2+1, 3+3+1, 2+2+2+1, 7x1} = 8;
a(8) = #{8, 7+1, 6+2, 5+3, 5+2+1, 4+4, 4+3+1, 3+3+2, 3+3+1+1, 2+2+2+2, 2+2+2+1+1, 8x1} = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

For median instead of mode we have A053263.
The complement is counted by A240302.
The case where the maximum is the only mode is A362612.
A000041 counts integer partitions, strict A000009.
A362608 counts partitions with a unique mode, complement A362607.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes.
Cf. A002865, A008284, A098859, A237984, A275870, A325347, A362610.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] == Count[p, Max[p]]], {n, 0, z}] (* this sequence *)
Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240302 *)
(* Clark Kimberling, Apr 04 2014 *)
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0],
     If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1,
     If[k == -1, j, If[k == 0, 0, If[j > k, 0, k]]]], {j, 1, n/i}]]];
a[n_] := PartitionsP[n] - b[n, n, -1];
a /@ Range[0, 70] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz in A240302 *)
Table[Length[Select[IntegerPartitions[n],MemberQ[Commonest[#],Max[#]]&]],{n,0,30}] (* Gus Wiseman, May 07 2023 *)

{ my(N=53, x='x+O('x^N));
my(gf=1+sum(i=1,N,sum(j=1,floor(N/i),x^(i*j)*prod(k=1,i-1,(1-x^(k*(j+1)))/(1-x^k)))));
Vec(gf) } \\ John Tyler Rascoe, Mar 09 2024

a(n) = p(n,0,1,1) with p(n,i,j,k) = if k<=n then p(n-k,i,j+1,k) +p(n,max(i,j),1,k+1) else (if j0 then 0 else 1).

a(n) + A240302(n) = A000041(n). - Clark Kimberling, Apr 04 2014.

G.f.: 1 + Sum_{i, j>0} x^(i*j) * Product_{k=1..i-1} ((1 - x^(k*(j+1)))/(1 - x^k)). - John Tyler Rascoe, Mar 09 2024

A053262 Coefficients of the 5th-order mock theta function chi_0(q).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 7, 9, 7, 12, 11, 13, 13, 17, 15, 21, 20, 24, 24, 29, 28, 36, 35, 40, 42, 50, 48, 58, 58, 67, 70, 80, 79, 93, 95, 106, 111, 125, 127, 145, 149, 166, 172, 191, 196, 222, 229, 250, 262, 289, 298, 330, 343, 373, 391, 427, 442, 486

Offset: 0

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Author

Dean Hickerson, Dec 19 1999

Keywords

nonn
easy

Comments

The rank of a partition is its largest part minus the number of parts.

References

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 23, 25.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (corrected and extended previous b-file from G. C. Greubel)

George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.

George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.

George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053263, A053264, A053265, A053266, A053267.

Programs

Mathematica

1+Series[Sum[q^(2n+1)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}] nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)/Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[nmax/2]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Formula

G.f.: chi_0(q) = Sum_{n>=0} q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))).

G.f.: chi_0(q) = 1 + Sum_{n>=0} q^(2n+1)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).

a(n) is the number of partitions of 5n with rank == 1 (mod 5) minus number with rank == 0 (mod 5).

a(n) is the number of partitions of n with unique smallest part and all other parts <= twice the smallest part.

a(n) is the number of partitions where the largest part is odd and all other parts are greater than half of the largest part. - N. Sato, Jan 21 2010

a(n) ~ exp(Pi*sqrt(2*n/15)) / sqrt((5 + sqrt(5))*n). - Vaclav Kotesovec, Jun 12 2019

A053264 Coefficients of the '5th-order' mock theta function F_0(q).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 10, 11, 11, 13, 14, 15, 17, 18, 19, 22, 24, 25, 28, 30, 32, 36, 39, 41, 45, 49, 52, 57, 61, 65, 71, 76, 81, 88, 94, 100, 109, 116, 123, 133, 142, 151, 163, 174, 184, 198, 211, 224, 240, 255, 271, 290

Offset: 0

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Author

Dean Hickerson, Dec 19 1999

Keywords

nonn
easy

References

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 22, 23, 25.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.

George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.

George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053265, A053266, A053267.

Programs

Mathematica

Series[Sum[q^(2n^2)/Product[1-q^(2k+1), {k, 0, n-1}], {n, 0, 7}], {q, 0, 100}] nmax = 100; CoefficientList[Series[Sum[x^(2*k^2) / Product[1-x^(2*j+1), {j, 0, k-1}], {k, 0, Floor[Sqrt[nmax/2]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Formula

G.f.: F_0(q) = Sum_{n>=0} q^(2n^2)/((1-q)(1-q^3)...(1-q^(2n-1))).

a(n) is the number of partitions of n into odd parts, each of which occurs at least twice, such that if k occurs then all smaller positive odd numbers occur.

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(3/2)*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053267 Coefficients of the '5th-order' mock theta function Psi(q).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 4, 4, 5, 5, 7, 6, 8, 8, 9, 9, 12, 11, 14, 14, 16, 16, 20, 19, 23, 24, 27, 27, 32, 32, 37, 38, 43, 44, 51, 51, 58, 61, 67, 69, 78, 80, 89, 93, 102, 106, 118, 121, 134, 140, 153, 159, 175, 181, 198, 207, 224, 234, 256, 265, 288

Offset: 0

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History

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Author

Dean Hickerson, Dec 19 1999

Keywords

nonn
easy

References

Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266.

Programs

Mathematica

Series[Sum[q^(5n^2)/Product[1-q^Abs[5k+2], {k, -n, n}], {n, 0, 4}], {q, 0, 100}]-1 nmax = 100; CoefficientList[Series[-1 + Sum[x^(5*k^2)/ Product[1-x^Abs[5*j+2], {j, -k, k}], {k, 0, Floor[Sqrt[nmax/5]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Formula

G.f.: Psi(q) = -1 + Sum_{n>=0} q^(5n^2)/((1-q^2)(1-q^3)(1-q^7)(1-q^8)...(1-q^(5n+2))).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(3/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053260 Coefficients of the '5th-order' mock theta function psi_0(q).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 4, 4, 4, 5, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 9, 8, 9, 10, 9, 11, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 29, 30, 32, 32, 34, 36, 36, 39, 40, 41, 44, 45, 46

Offset: 0

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History

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Author

Dean Hickerson, Dec 19 1999

Keywords

nonn
easy

Comments

Number of partitions of n such that each part occurs at most twice, the largest part is unique and if k occurs as a part then all smaller positive integers occur.

Strongly unimodal compositions with first part 1 and each up-step is by at most 1 (left-smoothness); with this interpretation one should set a(0)=1; see example. Replacing "strongly" by "weakly" in the condition gives A001524. Dropping the requirement of unimodality gives A005169. [Joerg Arndt, Dec 09 2012]

Examples

From _Joerg Arndt_, Dec 09 2012: (Start) The a(42)=8 strongly unimodal left-smooth compositions are [ #] composition [ 1] [ 1 2 3 4 5 6 7 5 4 3 2 ] [ 2] [ 1 2 3 4 5 6 7 6 4 3 1 ] [ 3] [ 1 2 3 4 5 6 7 6 5 2 1 ] [ 4] [ 1 2 3 4 5 6 7 6 5 3 ] [ 5] [ 1 2 3 4 5 6 7 8 3 2 1 ] [ 6] [ 1 2 3 4 5 6 7 8 4 2 ] [ 7] [ 1 2 3 4 5 6 7 8 5 1 ] [ 8] [ 1 2 3 4 5 6 7 8 6 ] (End)

References

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.

Links

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

George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.

George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.

George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i-1)))) end: a:= proc(n) local h, k, m, r; m, r:= floor((sqrt(n*8+1)-1)/2), 0; for k from m by -1 do h:= k*(k+1); if h<=n then break fi; r:= r+b(n-h/2, k-1) od: r end: seq(a(n), n=0..100); # Alois P. Heinz, Aug 02 2013

Mathematica

Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}], {n, 0, 12}], {q, 0, 100}] b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-1] ] ]]; a[n_] := Module[{h, k, m, r}, {m, r} = {Floor[(Sqrt[n*8+1]-1)/2], 0}; For[k = m, True, k--, h = k*(k+1); If[h <= n, Break[]]; r = r + b[n-h/2, k-1]]; r]; Table[ a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2015, after Alois P. Heinz *)

PARI

N = 66; x = 'x + O('x^N); gf = sum(n=1,N, x^(n*(n+1)/2) * prod(k=1,n-1,1+x^k) ) + 'c0; v = Vec(gf); v[1]-='c0; v /* Joerg Arndt, Apr 21 2013 */

Formula

G.f.: psi_0(q) = Sum_{n>=0} q^((n+1)*(n+2)/2) * (1+q)*(1+q^2)*...*(1+q^n).

a(n) ~ exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A053265 Coefficients of the '5th-order' mock theta function F_1(q).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 20, 21, 24, 26, 28, 31, 34, 37, 40, 44, 47, 51, 56, 60, 65, 71, 76, 82, 89, 95, 103, 111, 119, 128, 138, 148, 158, 171, 182, 195, 210, 223, 239, 256, 273, 292, 312, 332, 354, 378, 402, 428

Offset: 0

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Author

Dean Hickerson, Dec 19 1999

Keywords

nonn
easy

References

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 22, 25.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.

George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.

George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053266, A053267.

Programs

Mathematica

Series[Sum[q^(2n(n+1))/Product[1-q^(2k+1), {k, 0, n}], {n, 0, 6}], {q, 0, 100}] nmax = 100; CoefficientList[Series[Sum[x^(2*k*(k+1)) / Product[1-x^(2*j+1), {j, 0, k}], {k, 0, Floor[Sqrt[nmax/2]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Formula

G.f.: F_1(q) = Sum_{n>=0} q^(2n(n+1))/((1-q)(1-q^3)...(1-q^(2n+1))).

a(n) ~ sqrt(phi) * exp(Pi*sqrt(2*n/15)) / (2^(3/2)*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

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A238478 Number of partitions of n whose median is a part.

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A053266 Coefficients of the '5th-order' mock theta function Phi(q).

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A237821 Number of partitions of n such that 2*(least part) <= greatest part.

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A053256 Coefficients of the '5th-order' mock theta function f_0(q).

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A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

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A053262 Coefficients of the 5th-order mock theta function chi_0(q).

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A053264 Coefficients of the '5th-order' mock theta function F_0(q).

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