A053993 -id:A053993 - cp's OEIS frontend

A227551 Number T(n,k) of partitions of n into distinct parts with boundary size k; triangle T(n,k), n>=0, 0<=k<=A227568(n), read by rows.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 5, 2, 0, 1, 5, 4, 0, 1, 5, 6, 0, 1, 6, 7, 1, 0, 1, 6, 10, 1, 0, 1, 7, 11, 3, 0, 1, 9, 13, 4, 0, 1, 7, 18, 6, 0, 1, 8, 20, 9, 0, 1, 10, 21, 14, 0, 1, 9, 27, 16, 1, 0, 1, 10, 29, 22, 2

Offset: 0

Alois P. Heinz, Jul 16 2013

The boundary size is the number of parts having fewer than two neighbors.

			T(12,1) = 1: [12].
T(12,2) = 6: [1,11], [2,10], [3,4,5], [3,9], [4,8], [5,7].
T(12,3) = 7: [1,2,3,6], [1,2,9], [1,3,8], [1,4,7], [1,5,6], [2,3,7], [2,4,6].
T(12,4) = 1: [1,2,4,5].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 3;
  0, 1, 3, 1;
  0, 1, 3, 2;
  0, 1, 5, 2;
  0, 1, 5, 4;
  0, 1, 5, 6;
  0, 1, 6, 7, 1;

Alois P. Heinz, Rows n = 0..600, flattened

Columns k=0-10 give: A000007, A057427, A227559, A227560, A227561, A227562, A227563, A227564, A227565, A227566, A227567.
Row sums give: A000009.
Last elements of rows give: A227552.
Cf. A227345 (a version with trailing zeros), A053993, A201077, A227568, A224878 (one part of size 0 allowed).

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),
      expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
      `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
    end:
T:= n-> (p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..30);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t > 1, x, 1], Expand[If[i < 1, 0, If[t > 1, x, 1]*b[n, i - 1, Quotient[t, 2]] + If[i > n, 0, If[t == 2, x, 1]*b[n - i, i - 1, Quotient[t, 2] + 2]]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 12 2016, after Alois P. Heinz *)

A097242 Expansion of q-series 1 / (q^2, q^3, q^9, q^10; q^12)_infinity.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 3, 3, 3, 5, 4, 6, 7, 7, 8, 11, 10, 13, 15, 16, 18, 23, 22, 27, 31, 33, 37, 45, 45, 53, 60, 64, 71, 84, 86, 99, 111, 119, 131, 151, 157, 178, 198, 212, 233, 264, 277, 310, 342, 367, 401, 449, 474, 525, 576, 618, 673, 746, 790, 869, 949, 1017, 1104

Offset: 0

Michael Somos, Aug 02 2004

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of n into odd parts in which every part occurs at least twice. Example: a(9)=3 because we have [3,3,3], [3,3,1,1,1] and [1,1,1,1,1,1,1,1,1]. - Vladeta Jovovic, Jan 16 2005
Also equal to the number of partitions of n into distinct parts not congruent to 1 or 5 modulo 6.   Example: a(9) = 3, the relevant partitions being [9], [6,3], and [4,3,2]. - Jeremy Lovejoy, Jun 21 2020
From Joerg Arndt, Jun 21 2020: (Start)
a(n) is the number of partitions with parts == { 2, 3, 9, 10 } (mod 12).
a(n) is the number of overpartitions with non-overlined parts == 2 (mod 4) and overlined parts == 3 (mod 6); same as the number of partitions with parts == 2 (mod 4) and distinct parts == 3 (mod 6).  (End)

			G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 3*x^9 + 3*x^10 + 3*x^11 + ...
G.f. = 1/q + q^47 + q^71 + q^95 + q^119 + 2*q^143 + q^167 + 2*q^191 + 3*q^215 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
K. Alladi, G.E. Andrews, and B. Gordon, Refinements and generalizations of Capparelli's conjecture on partitions, Journal of Algebra, 174 (1995), 636-658.
J. Dousse and J. Lovejoy, Generalizations of Capparelli's identity, Bulletin of the London Mathematical Society, 51 (2019), 193-206.
J. Lovejoy, Asymmetric generalizations of Schur's theorem, in: Andrews G., Garvan F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham.
A. V. Sills, On series expansion of Capparelli's infinite product, Adv. in Appl. Math., 33 (2004), pp. 397-408.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053993.

g:=product(1+x^(4*j-2)/(1-x^(2*j-1)),j=1..20): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..65); # Emeric Deutsch, Feb 23 2006

nmax = 100; CoefficientList[Series[Product[1/((1 - x^(12*k + 2)) * (1 - x^(12*k + 3)) * (1 - x^(12*k + 9)) * (1 - x^(12*k + 10))), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] QPochhammer[ -x^3, x^6], {x, 0, n}]; (* Michael Somos, Jan 09 2017 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - x^k * [0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0][(k-1)%12 + 1], 1 + x * O(x^n)), n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))}; /* Michael Somos, Oct 17 2006 */

G.f.: Product_{k>0} (1 + x^(6*k - 3)) / (1 - x^(4*k - 2)).
G.f.: 1 / (Product_{k>=0} (1 - x^(12*k + 2)) * (1 - x^(12*k + 3)) * (1 - x^(12*k + 9)) * (1 - x^(12*k + 10))).
Expansion of chi(x^3) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Oct 17 2006
Expansion of q^(1/24) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, ...].
a(n) ~ Pi*BesselI(1, sqrt(24*n-1)*Pi/(6*sqrt(3))) / sqrt(3*(24*n-1)/2) ~ exp(Pi*sqrt(2*n)/3) / (2^(7/4) * sqrt(3) * n^(3/4)) * (1 - (9/(8*Pi) + Pi/72)/sqrt(2*n) + (5/128 - 135/(256*Pi^2) + Pi^2/20736)/n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 09 2017

A186084 Triangle T(n,k) read by rows: number of 1-dimensional sandpiles (see A186085) with n grains and base length k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 6, 5, 1, 0, 0, 0, 0, 1, 4, 10, 6, 1, 0, 0, 0, 0, 0, 3, 10, 15, 7, 1, 0, 0, 0, 0, 0, 2, 8, 20, 21, 8, 1, 0, 0, 0, 0, 0, 1, 7, 19, 35, 28, 9, 1, 0, 0, 0, 0, 0, 0, 5, 18, 40, 56, 36, 10, 1, 0, 0, 0, 0, 0, 0, 3, 16, 41, 76, 84, 45, 11, 1, 0, 0, 0, 0, 0, 0, 1, 12, 41, 86, 133, 120, 55, 12, 1

Offset: 0

Joerg Arndt, Feb 13 2011

Compositions of n into k nonzero parts such that the first and last parts are 1 and the absolute difference between consecutive parts is <=1.
Row sums are A186085.
Column sums are the Motzkin numbers (A001006).
First nonzero entry in row n appears in column A055086(n).
From Joerg Arndt, Nov 06 2012: (Start)
The transposed triangle (with zeros omitted) is A129181.
For large k, the columns end in reverse([1, 1, 3, 5, 9, 14, 24, 35, ...]) for k even (cf. A053993) and reverse([1, 2, 3, 6, 10, 16, 26, 40, 60, 90, ...]) for k odd (cf. A201077).
The diagonals below the main diagonal are (apart from leading zeros), n, n*(n+1)/2, n*(n+1)*(n+2)/6, and the e-th diagonal appears to have a g.f. of the form f(x)/(1-x)^e.
(End)

			Triangle begins:
1;
0,1;
0,0,1;
0,0,1,1;
0,0,0,2,1;
0,0,0,1,3,1;
0,0,0,0,3,4,1;
0,0,0,0,1,6,5,1;
0,0,0,0,1,4,10,6,1;
0,0,0,0,0,3,10,15,7,1;
0,0,0,0,0,2,8,20,21,8,1;
0,0,0,0,0,1,7,19,35,28,9,1;

Alois P. Heinz, Rows n = 0..140, flattened
Joerg Arndt, the first 36 rows as Pari script.

Cf. A186085 (sandpiles with n grains, row sums), A001006 (Motzkin numbers, column sums), A055086.

Cf. A186505 (antidiagonal sums).

b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),
      `if`(n<0 or i<1, 0, expand(x*add(b(n-i, i+j), j=-1..1)) ))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 24 2013

b[n_, i_] := b[n, i] = If[n == 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, Expand[ x*Sum[b[n-i, i+j], {j, -1, 1}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

{T(n,k)=local(A=1+x*y);for(i=1,n,A=1/(1-x*y-x^3*y^2*subst(A,y,x*y+x*O(x^n))));polcoeff(polcoeff(A,n,x),k,y)}
/* Paul D. Hanna */

G.f. A(x,y) satisfies:  A(x,y) = 1/(1 - x*y - x^3*y^2*A(x, x*y) ). [Paul D. Hanna, Feb 22 2011]
G.f.: (formatting to make the structure apparent)
A(x,y) = 1 /
(1 - x^1*y / (1 - x^2*y / (1 - x^5*y^2 /
(1 - x^3*y / (1 - x^4*y / (1 - x^9*y^2 /
(1 - x^5*y / (1 - x^6*y / (1 - x^13*y^2 /
(1 - x^7*y / (1 - x^8*y / (1 - x^17*y^2 / (1 -...)))))))))))))
(continued fraction). [Paul D. Hanna]
G.f.: A(x,y) = 1/(1-x*y - x^3*y^2/(1-x^2*y - x^5*y^2/(1-x^3*y - x^7*y^2/(1 -...)))) (continued fraction). [Paul D. Hanna]

A227552 Number of partitions of n into distinct parts with maximal boundary size.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 2, 2, 4, 6, 1, 1, 3, 4, 6, 9, 14, 1, 2, 3, 5, 8, 11, 17, 24, 1, 1, 3, 5, 8, 11, 18, 24, 35, 49, 1, 2, 3, 6, 9, 14, 21, 30, 42, 60, 81, 1, 1, 3, 5, 9, 13, 21, 29, 43, 60, 84, 113, 156, 1, 2, 3, 6, 10, 15, 24, 35, 50, 71, 99, 134, 184, 246

Offset: 0

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having less than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Last elements of rows of A227551.
Last nonzero elements of rows of A227345.
Cf. A053993, A201077.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),
      expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
      `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
    end:
a:= n-> (p->coeff(p, x, degree(p)))(b(n$2, 0)):
seq(a(n), n=0..100);

b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t>1, x, 1], Expand[If[i<1, 0, If[t>1, x, 1]*b[n, i-1, Quotient[t, 2]] + If[i>n, 0, If[t==2, x, 1] * b[n-i, i-1, Quotient[t, 2]+2]]]]]; a[n_] := Function [p, Coefficient[p, x, Exponent[p, x]]][b[n, n, 0]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

a(n) = A227551(n,A227568(n)).

cp's OEIS Frontend

A227551 Number T(n,k) of partitions of n into distinct parts with boundary size k; triangle T(n,k), n>=0, 0<=k<=A227568(n), read by rows.

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A097242 Expansion of q-series 1 / (q^2, q^3, q^9, q^10; q^12)_infinity.

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A186084 Triangle T(n,k) read by rows: number of 1-dimensional sandpiles (see A186085) with n grains and base length k.

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A227552 Number of partitions of n into distinct parts with maximal boundary size.

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A247661 The number phi_4(n) of Frobenius partitions that allow up to 4 repetitions of an integer in a row.

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A247662 The number phi_5(n) of Frobenius partitions that allow up to 5 repetitions of an integer in a row.

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