A000716 -id:A000716 - cp's OEIS frontend

A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 2, 4, 2, 2, 3, 5, 3, 4, 3, 6, 4, 4, 3, 5, 8, 5, 6, 6, 5, 10, 6, 8, 6, 5, 7, 12, 8, 10, 9, 10, 7, 15, 10, 12, 12, 10, 7, 11, 18, 12, 16, 15, 15, 14, 11, 22, 15, 20, 18, 20, 14, 11, 15

Offset: 0

Gary W. Adamson, Mar 27 2010

Row sums = A000041, the partition numbers.
The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.
k-th triangle in the infinite set can be defined as having the sequence:
"Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].
Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.
Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.
All triangles in the infinite set have row sums = A000041.
The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:
...
k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies
.....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =
.....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...
...
k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies
.....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get
.....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...
...
k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies
.....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get
.....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...
...
Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - Gus Wiseman, Oct 23 2023

			First few rows of the triangle =
1;
1;
1, 1;
2, 1;
2, 1, 2;
3, 2, 2;
4, 2, 2, 3;
5, 3, 4, 3;
6, 4, 4, 3, 5;
8, 5, 6, 6, 5;
10, 6, 8, 6, 5, 7;
12, 8, 10, 9, 10, 7;
15, 10, 12, 12, 10, 7, 11;
18, 12, 16, 15, 15, 14, 11;
22, 15, 20, 18, 20, 14, 11, 15;
...
From _Gus Wiseman_, Oct 23 2023: (Start)
Row n = 9 counts the following partitions:
  (9)          (72)        (54)       (63)      (81)
  (711)        (5211)      (522)      (6111)    (621)
  (531)        (3321)      (4311)     (432)     (441)
  (51111)      (321111)    (411111)   (42111)   (4221)
  (333)        (21111111)  (32211)    (3222)    (22221)
  (33111)                  (2211111)  (222111)
  (3111111)
  (111111111)
(End)

Cf. A000009, A173305, A174712, A174714, A174715.
Row sums are A000041.
The odd version is A365067.
The corresponding rank statistic is A366531, odd version A366528.
A000009 counts partitions into odd parts, ranks A066208.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A066967, A130780, A171966, A182616, A366527, A366533.

Table[Length[Select[IntegerPartitions[n],Total[Select[#,EvenQ]]==k&]],{n,0,15},{k,0,n,2}] (* Gus Wiseman, Oct 23 2023 *)

As infinite lower triangular matrices, A173305 * A174712.

T(n,k) = A000009(n-2k) * A000041(k). - Gus Wiseman, Oct 23 2023

A000991 Number of 3-line partitions of n.

Original entry on oeis.org

1, 1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, 1274, 1983, 3032, 4610, 6915, 10324, 15235, 22371, 32554, 47119, 67689, 96763, 137404, 194211, 272939, 381872, 531576, 736923, 1016904, 1397853, 1913561, 2610023, 3546507, 4802694, 6481101, 8718309, 11689929, 15627591, 20828892

Offset: 0

N. J. A. Sloane

nonn

Planar partitions into at most three rows. - Joerg Arndt, May 01 2013

Number of partitions of n where there is one sort of part 1, two sorts of part 2, and three sorts of every other part. - Joerg Arndt, Mar 15 2014

L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.8).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000 (first 1000 terms from Alois P. Heinz)
M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116.

A row of the array in A242641.

Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^2*(1-x^2)/(&*[1-x^j: j in [1..2*m]])^3 )); // G. C. Greubel, Dec 06 2018

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(min(i, 3)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 15 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[Min[i, 3]+j-1, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)
nmax = 40; CoefficientList[Series[(1-x)^2 * (1-x^2) * Product[1/(1-x^k)^3, {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Oct 28 2015 *)

x='x+O('x^66); Vec((1-x)^2*(1-x^2)/eta(x)^3) \\ Joerg Arndt, May 01 2013

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = (1-x)^2 * (1-x^2) / prod(1-x^j for j in (1..60))^3
s.coefficients()
# G. C. Greubel, Dec 06 2018

G.f.: (1-x)^2 * (1-x^2) / Product_(k>=1, 1-x^k )^3.
For n>=4, a(n) = A000716(n) - 2*A000716(n-1) + 2*A000716(n-3) - A000716(n-4). - Vaclav Kotesovec, Oct 28 2015
a(n) ~ Pi^3 * exp(Pi*sqrt(2*n)) / (16*n^3). - Vaclav Kotesovec, Oct 28 2015

G.f. corrected by Sean A. Irvine, Oct 19 2011
G.f. corrected by Joerg Arndt, May 01 2013
Prepended a(0)=1, added more terms, Joerg Arndt, May 01 2013

A341221 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^3.

Original entry on oeis.org

1, 6, 21, 59, 144, 321, 669, 1323, 2511, 4604, 8202, 14253, 24241, 40449, 66363, 107234, 170910, 269004, 418566, 644436, 982536, 1484482, 2223942, 3305484, 4876620, 7144455, 10398123, 15039564, 21624678, 30919323, 43973708, 62222844, 87619212, 122810585

Offset: 3

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Column k=3 of A060642.

Cf. A000041, A000716, A048574, A327381, A341222, A341223, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..36);  # Alois P. Heinz, Feb 07 2021

nmax = 36; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

a(n) ~ A000716(n). - Vaclav Kotesovec, Feb 20 2021

A355350 G.f. A(x,y) satisfies: xy = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 9, 6, 1, 0, 22, 27, 10, 1, 0, 51, 98, 66, 15, 1, 0, 108, 315, 340, 135, 21, 1, 0, 221, 918, 1495, 910, 246, 28, 1, 0, 429, 2492, 5838, 5070, 2086, 413, 36, 1, 0, 810, 6372, 20805, 24543, 14280, 4284, 652, 45, 1, 0, 1479, 15525, 68816, 106535, 83559, 35168, 8100, 981, 55, 1, 0, 2640, 36280, 213945, 423390, 432930, 243208, 78282, 14355, 1420, 66, 1

Offset: 0

Paul D. Hanna, Jun 29 2022

The term T(n,k) is found in row n and column k of this triangle, and can be used to derive the following sequences.
A355351(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).
A355352(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.
A355353(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.
A355354(n) = Sum_{k=0..n} T(n,k) * 4^k for n >= 0.
A355355(n) = Sum_{k=0..n} T(n,k) * 5^k for n >= 0.
A355356(n) = Sum_{k=0..floor(n/2)} T(n-k,k) for n >= 0 (antidiagonal sums).
A355357(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) for n >= 0.
A354658(n) = T(2*n,n) for n >= 0 (central terms of this triangle).
Conjectures:
(C.1) Column 1 equals A000716, the number of partitions into parts of 3 kinds;
(C.2) Column 2 equals A023005, the number of partitions into parts of 6 kinds.

			G.f.: A(x,y) = 1 + x*y + x^2*(3*y + y^2) + x^3*(9*y + 6*y^2 + y^3) + x^4*(22*y + 27*y^2 + 10*y^3 + y^4) + x^5*(51*y + 98*y^2 + 66*y^3 + 15*y^4 + y^5) + x^6*(108*y + 315*y^2 + 340*y^3 + 135*y^4 + 21*y^5 + y^6) + x^7*(221*y + 918*y^2 + 1495*y^3 + 910*y^4 + 246*y^5 + 28*y^6 + y^7) + x^8*(429*y + 2492*y^2 + 5838*y^3 + 5070*y^4 + 2086*y^5 + 413*y^6 + 36*y^7 + y^8) + x^9*(810*y + 6372*y^2 + 20805*y^3 + 24543*y^4 + 14280*y^5 + 4284*y^6 + 652*y^7 + 45*y^8 + y^9) + x^10*(1479*y + 15525*y^2 + 68816*y^3 + 106535*y^4 + 83559*y^5 + 35168*y^6 + 8100*y^7 + 981*y^8 + 55*y^9 + y^10) + ...
where
x*y = ... - x^10/A(x,y)^5 + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...
also, given P(x) is the partition function (A000041),
x*y*P(x) = (1 - x*A(x,y))*(1 - 1/A(x,y)) * (1 - x^2*A(x,y))*(1 - x/A(x,y)) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y)) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y)) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y)) * ...
TRIANGLE.
The triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for k = 0..n in row n, begins:
n=0: [1];
n=1: [0, 1];
n=2: [0, 3, 1];
n=3: [0, 9, 6, 1];
n=4: [0, 22, 27, 10, 1];
n=5: [0, 51, 98, 66, 15, 1];
n=6: [0, 108, 315, 340, 135, 21, 1];
n=7: [0, 221, 918, 1495, 910, 246, 28, 1];
n=8: [0, 429, 2492, 5838, 5070, 2086, 413, 36, 1];
n=9: [0, 810, 6372, 20805, 24543, 14280, 4284, 652, 45, 1];
n=10: [0, 1479, 15525, 68816, 106535, 83559, 35168, 8100, 981, 55, 1];
n=11: [0, 2640, 36280, 213945, 423390, 432930, 243208, 78282, 14355, 1420, 66, 1];
n=12: [0, 4599, 81816, 630890, 1563705, 2033244, 1472261, 629280, 160965, 24145, 1991, 78, 1];
...
in which column 1 appears to equal A000716, the coefficients in P(x)^3,
and column 2 appears to equal A023005, the coefficients in P(x)^6,
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + ... + A000041(n)*x^n + ...
Also, the power series expansions of P(x)^3 and P(x)^6 begin
P(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 + 429*x^7 + 810*x^8 + 1479*x^9 + 2640*x^10 + ... + A000716(n)*x^n + ...
P(x)^6 = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2492*x^6 + 6372*x^7 + 15525*x^8 + 36280*x^9 + 81816*x^10 + ... + A023005(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..5150

Cf. A355351 (row sums), A355352, A355353, A355354, A355355.
Cf. A355356, A355357, A354658 (central terms).
Cf. A354645, A354650 (related table), A000041, A000716, A023005.

{T(n,k) = my(A=[1,y],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));polcoeff(A[n+1],k,y)}
for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))

G.f. A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*y^k satisfies:
(1) x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) x*y*P(x) = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

A339318 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^3.

Original entry on oeis.org

1, 3, 3, 9, 3, 12, 3, 22, 9, 12, 3, 39, 3, 12, 12, 51, 3, 39, 3, 39, 12, 12, 3, 105, 9, 12, 22, 39, 3, 57, 3, 108, 12, 12, 12, 135, 3, 12, 12, 105, 3, 57, 3, 39, 39, 12, 3, 258, 9, 39, 12, 39, 3, 105, 12, 105, 12, 12, 3, 201, 3, 12, 39, 221, 12, 57, 3, 39, 12, 57

Offset: 1

Ilya Gutkovskiy, Nov 30 2020

nonn

Number of factorizations of n into factors (greater than 1) of 3 kinds.

			From _Antti Karttunen_, Dec 15 2021: (Start)
For n = 8, A001055(8) = 3, as it has three ordinary factorizations: (8), (4*2), (2*2*2). When allowing each of the factors appear in three different guises (here indicated with a subscript), and where neither the order of factors nor their subscripts matter, we get the following 22 different factorizations:
  (8_3), (8_2), (8_1),
  (4_3 * 2_3), (4_3 * 2_2), (4_3 * 2_1),
  (4_2 * 2_3), (4_2 * 2_2), (4_2 * 2_1),
  (4_1 * 2_3), (4_1 * 2_2), (4_1 * 2_1),
  (2_3 * 2_3 * 2_3), (2_3 * 2_3 * 2_2), (2_3 * 2_3 * 2_1),
  (2_3 * 2_2 * 2_2), (2_3 * 2_2 * 2_1), (2_3 * 2_1 * 2_1),
  (2_2 * 2_2 * 2_2), (2_2 * 2_2 * 2_1), (2_2 * 2_1 * 2_1),
  (2_1 * 2_1 * 2_1),
therefore a(8) = 22. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000716, A001055, A301830, A339319, A339320, A339321, A339322, A339323, A339324.

A339318list(n) = MultEulerT(vector(n, i, 3)); \\ Antti Karttunen, Jan 21 2022, using Andrew Howroyd's program given in A301830.

a(p^k) = A000716(k) for prime p.

A356500 Coefficients T(n,k) of x^ny^k in A(x,y) for n >= 0, k = 0..3n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 6, 0, 0, 0, 28, 0, 0, 0, 22, 0, 3, 0, 0, 0, 84, 0, 0, 0, 219, 0, 0, 0, 140, 0, 0, 0, 0, 135, 0, 0, 0, 981, 0, 0, 0, 1807, 0, 0, 0, 969, 0, 0, 0, 120, 0, 0, 0, 2568, 0, 0, 0, 10764, 0, 0, 0, 15368, 0, 0, 0, 7084, 0, 0, 54, 0, 0, 0, 4284, 0, 0, 0, 38896, 0, 0, 0, 114240, 0, 0, 0, 133266, 0, 0, 0, 53820, 0, 9, 0, 0, 0, 4662, 0, 0, 0, 94390, 0, 0, 0, 525980, 0, 0, 0, 1187433, 0, 0, 0, 1171390, 0, 0, 0, 420732

Offset: 0

Paul D. Hanna, Aug 09 2022

T(n, 3*n+1) = [x^n*y^(3*n+1)] A(x,y) = binomial(4*n, n)/(3*n + 1) = A002293(n) for n >= 0, where g.f. G(x) of A002293 satisfies: G(x) = 1 + x*G(x)^4.
T(4*n, 1) = A000716(n) for n >= 0 (nonzero terms in column 1).
T(4*n+3, 2) = [x^(4*n+3)*y^2] A(x,y) = 2 * A354655(n+1) for n >= 0, where A354655 equals column 2 of triangle A354650.
T(4*n+2, 3) = [x^(4*n+2)*y^3] A(x,y) = 4 * A354656(n+1) for n >= 0, where A354656 equals column 3 of triangle A354650.
T(2*n, 2*n+1) = A356504(n), for n >= 0.
T(2*n+1, 2*n) = A356505(n) for n >= 0.
T(3*n, n+1) = A356506(n) for n >= 0.
T(3*n+1, n) = A355365(n) where A355365 is the central terms of A355360 = A355360(2*n,n).
Sum_{k=0..3*n+1} T(n, k) = A354248(n) for n >= 0 (row sums).
Sum_{k=0..3*n+1} T(n, k) * 2^k = A356502(n) for n >= 0.
Sum_{k=0..3*n+1} T(n, k) * 3^k = A356503(n) for n >= 0.
Sum_{k=0..3*n+1} T(4*n+1-k, k) = A355872(n+1) for n >= 0 (nonzero antidiagonal sums).
SPECIFIC VALUES.
(V.1) 1 = A(x,y) at x = -exp(-Pi) and y = Pi^(1/4)/gamma(3/4).
(V.2) 1 = A(x,y) at x = -exp(-2*Pi) and y = Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) 1 = A(x,y) at x = -exp(-3*Pi) and y = Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) 1 = A(x,y) at x = -exp(-4*Pi) and y = Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) 1 = A(x,y) at x = -exp(-sqrt(3)*Pi) and y = gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

			G.f.: A(x,y) = y + x*(1 + y^4) + x^2*(4*y^3 + 4*y^7) + x^3*(6*y^2 + 28*y^6 + 22*y^10) + x^4*(3*y + 84*y^5 + 219*y^9 + 140*y^13) + x^5*(135*y^4 + 981*y^8 + 1807*y^12 + 969*y^16) + x^6*(120*y^3 + 2568*y^7 + 10764*y^11 + 15368*y^15 + 7084*y^19) + x^7*(54*y^2 + 4284*y^6 + 38896*y^10 + 114240*y^14 + 133266*y^18 + 53820*y^22) + x^8*(9*y + 4662*y^5 + 94390*y^9 + 525980*y^13 + 1187433*y^17 + 1171390*y^21 + 420732*y^25) + x^9*(3250*y^4 + 160965*y^8 + 1670942*y^12 + 6640711*y^16 + 12167001*y^20 + 10399545*y^24 + 3362260*y^28) + ...
such that A = A(x,y) satisfies
y = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...
This irregular triangle of coefficients T(n,k) of x^n*y^k in A(x,y) for n >= 0, k = 0..3*n+1, begins:
  n = 0: [0, 1];
  n = 1: [1, 0, 0, 0, 1];
  n = 2: [0, 0, 0, 4, 0, 0, 0, 4];
  n = 3: [0, 0, 6, 0, 0, 0, 28, 0, 0, 0, 22];
  n = 4: [0, 3, 0, 0, 0, 84, 0, 0, 0, 219, 0, 0, 0, 140];
  n = 5: [0, 0, 0, 0, 135, 0, 0, 0, 981, 0, 0, 0, 1807, 0, 0, 0, 969];
  n = 6: [0, 0, 0, 120, 0, 0, 0, 2568, 0, 0, 0, 10764, 0, 0, 0, 15368, 0, 0, 0, 7084];
  n = 7: [0, 0, 54, 0, 0, 0, 4284, 0, 0, 0, 38896, 0, 0, 0, 114240, 0, 0, 0, 133266, 0, 0, 0, 53820];
  n = 8: [0, 9, 0, 0, 0, 4662, 0, 0, 0, 94390, 0, 0, 0, 525980, 0, 0, 0, 1187433, 0, 0, 0, 1171390, 0, 0, 0, 420732];
  n = 9: [0, 0, 0, 0, 3250, 0, 0, 0, 160965, 0, 0, 0, 1670942, 0, 0, 0, 6640711, 0, 0, 0, 12167001, 0, 0, 0, 10399545, 0, 0, 0, 3362260];
  ...
Reading this triangle by nonzero antidiagonals [x^(4*n+1-k)*y^k] A(x,y) for n >= 0, k = 0..3*n+1, yields triangle A356501:
  [1, 1];
  [0, 3, 6, 4, 1];
  [0, 9, 54, 120, 135, 84, 28, 4];
  [0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22];
  [0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140];
  [0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969];
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..3926

Cf. A354248, A356502, A356503, A356504, A356505, A355872.
Cf. A354655, A354656, A355365, A000716, A002293.
Cf. related tables: A356501, A355350, A355360, A355870.

{T(n,k) = my(A=[y],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[n+1],k,y) }
for(n=0,9, for(k=0,3*n+1, print1(T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..3*n+1} T(n,k) * x^n * y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n+1)^2).
(2) y = A(x,y) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n-1)*A(x,y)^(2*n+1)) * (1 - x^(2*n-1)*A(x,y)^(2*n-3)), by the Jacobi triple product identity.
(3) y = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n+1)*A(x,y)^(2*n-1)) * (1 - x^(2*n-3)*A(x,y)^(2*n-1)), by the Jacobi triple product identity.
(4) y = A(x, F(x,y)) where F(x,y) = Sum_{n=-oo..+oo} (-x)^(n^2) * y^((n-1)^2).
(5) 1 = A(x, theta_4(x)) where theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2) is a Jacobi theta function.

A385001 Irregular triangle read by rows: T(n,k) is the number of partitions of n with k designated summands, n >= 0, 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 4, 1, 0, 7, 3, 0, 6, 9, 0, 12, 15, 1, 0, 8, 30, 3, 0, 15, 45, 9, 0, 13, 67, 22, 0, 18, 99, 42, 1, 0, 12, 135, 81, 3, 0, 28, 175, 140, 9, 0, 14, 231, 231, 22, 0, 24, 306, 351, 51, 0, 24, 354, 551, 97, 1, 0, 31, 465, 783, 188, 3, 0, 18, 540, 1134, 330, 9

Offset: 0

Omar E. Pol, Jul 17 2025

The divisor function sigma_1(n) = A000203(n) is also the number of partitions of n with only one designated summand, n >= 1.
When part i has multiplicity j > 0 exactly one part i is "designated".
The length of the row n is A002024(n+1) = 1 + A003056(n), hence the first positive element in column k is in the row A000217(k).
Alternating row sums give A329157.
Columns converge to A000716.
This triangle equals A060043 with reversed rows and an additional column 0.

			Triangle begins:
--------------------------------------------
   n\k:   0    1     2     3     4    5   6
--------------------------------------------
   0 |    1;
   1 |    0,   1;
   2 |    0,   3;
   3 |    0,   4,    1;
   4 |    0,   7,    3;
   5 |    0,   6,    9;
   6 |    0,  12,   15,    1;
   7 |    0,   8,   30,    3;
   8 |    0,  15,   45,    9;
   9 |    0,  13,   67,   22;
  10 |    0,  18,   99,   42,    1;
  11 |    0,  12,  135,   81,    3;
  12 |    0,  28,  175,  140,    9;
  13 |    0,  14,  231,  231,   22;
  14 |    0,  24,  306,  351,   51;
  15 |    0,  24,  354,  551,   97,   1;
  16 |    0,  31,  465,  783,  188,   3;
  17 |    0,  18,  540, 1134,  330,   9;
  18 |    0,  39,  681, 1546,  568,  22;
  19 |    0,  20,  765, 2142,  918,  51;
  20 |    0,  42,  945, 2835, 1452, 108;
  21 |    0,  32, 1040, 3758, 2233, 208,  1;
  ...
For n = 6 and k = 1 there are 12 partitions of 6 with only one designated summand as shown below:
   6'
   3'+ 3
   3 + 3'
   2'+ 2 + 2
   2 + 2'+ 2
   2 + 2 + 2'
   1'+ 1 + 1 + 1 + 1 + 1
   1 + 1'+ 1 + 1 + 1 + 1
   1 + 1 + 1'+ 1 + 1 + 1
   1 + 1 + 1 + 1'+ 1 + 1
   1 + 1 + 1 + 1 + 1'+ 1
   1 + 1 + 1 + 1 + 1 + 1'
So T(6,1) = 12, the same as A000203(6) = 12.
.
For n = 6 and k = 2 there are 15 partitions of 6 with two designated summands as shown below:
   5'+ 1'
   4'+ 2'
   4'+ 1'+ 1
   4'+ 1 + 1'
   3'+ 1'+ 1 + 1
   3'+ 1 + 1'+ 1
   3'+ 1 + 1 + 1'
   2'+ 2 + 1'+ 1
   2'+ 2 + 1 + 1'
   2 + 2'+ 1'+ 1
   2 + 2'+ 1 + 1'
   2'+ 1'+ 1 + 1 + 1
   2'+ 1 + 1'+ 1 + 1
   2'+ 1 + 1 + 1'+ 1
   2'+ 1 + 1 + 1 + 1'
So T(6,2) = 15, the same as A002127(6) = 15.
.
For n = 6 and k = 3 there is only one partition of 6 with three designated summands as shown below:
   3'+ 2'+ 1'
So T(6,3) = 1, the same as A002128(6) = 1.
There are 28 partitions of 6 with designated summands, so A077285(6) = 28.
.

Alois P. Heinz, Rows n = 0..1000, flattened

Columns: A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5), A384926 (k=6).
Row sums give A077285.
Cf. A000217, A002024, A003056, A060043, A196020, A252117, A329157, A365434, A384999.
Cf. A000096, A000716, A116608, A293421, A384998, A384999.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 18 2025

From Alois P. Heinz, Jul 18 2025: (Start)
Sum_{k>=1} k * T(n,k) = A293421(n).
T(A000096(n),n) = A000716(n). (End)
G.f.: Product_{i>0} 1 + (y*x^i)/(1 - x^i)^2. - John Tyler Rascoe, Jul 23 2025
Conjecture: For fixed k >= 1, Sum_{j=1..n} T(j,k) ~ Pi^(2*k) * n^(2*k) / ((2*k)! * (2*k+1)!). - Vaclav Kotesovec, Aug 01 2025

A355360 G.f. A(x,y) satisfies: xyA(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 0, 9, 12, 5, 0, 22, 54, 46, 14, 0, 51, 196, 282, 175, 42, 0, 108, 630, 1360, 1365, 666, 132, 0, 221, 1836, 5635, 8190, 6321, 2541, 429, 0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430, 0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862, 0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796

Offset: 0

Paul D. Hanna, Jul 19 2022

The main diagonal equals A000108, the Catalan numbers.
Conjectures.
(C.1) Column 1 equals A000716, the number of partitions into parts of 3 kinds.
(C.2) Column 2 equals twice A023005, the number of partitions into parts of 6 kinds.
The term T(n,k) is found in row n and column k of this triangle, and can be used to derive the following sequences.
A355361(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).
A355362(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.
A355363(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.
A355364(n) = Sum_{k=0..floor(n/2)} T(n-k,k) for n >= 0 (antidiagonal sums).
A355365(n) = T(2*n,n) for n >= 0 (central terms of this triangle).

			G.f.: A(x,y) = 1 + x*y + x^2*(3*y + 2*y^2) + x^3*(9*y + 12*y^2 + 5*y^3) + x^4*(22*y + 54*y^2 + 46*y^3 + 14*y^4) + x^5*(51*y + 196*y^2 + 282*y^3 + 175*y^4 + 42*y^5) + x^6*(108*y + 630*y^2 + 1360*y^3 + 1365*y^4 + 666*y^5 + 132*y^6) + x^7*(221*y + 1836*y^2 + 5635*y^3 + 8190*y^4 + 6321*y^5 + 2541*y^6 + 429*y^7) + x^8*(429*y + 4984*y^2 + 20850*y^3 + 41405*y^4 + 45326*y^5 + 28448*y^6 + 9724*y^7 + 1430*y^8) + x^9*(810*y + 12744*y^2 + 70737*y^3 + 184527*y^4 + 270060*y^5 + 237209*y^6 + 125532*y^7 + 37323*y^8 + 4862*y^9) + x^10*(1479*y + 31050*y^2 + 223652*y^3 + 745745*y^4 + 1404102*y^5 + 1625932*y^6 + 1193116*y^7 + 546039*y^8 + 143650*y^9 + 16796*y^10) + ...
where
x*y*A(x) = ... - x^10/A(x,y)^5 + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...
also,
x*y*A(x)*P(x) = (1 - x*A(x,y))*(1 - 1/A(x,y)) * (1 - x^2*A(x,y))*(1 - x/A(x,y)) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y)) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y)) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y)) * ...
TRIANGLE.
The triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for k = 0..n in row n, begins:
n=0: [1];
n=1: [0, 1];
n=2: [0, 3, 2];
n=3: [0, 9, 12, 5];
n=4: [0, 22, 54, 46, 14];
n=5: [0, 51, 196, 282, 175, 42];
n=6: [0, 108, 630, 1360, 1365, 666, 132];
n=7: [0, 221, 1836, 5635, 8190, 6321, 2541, 429];
n=8: [0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430];
n=9: [0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862];
n=10: [0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796];
n=11: [0, 2640, 72560, 667005, 2784110, 6565030, 9646462, 9242178, 5826171, 2349490, 554268, 58786];
n=12: [0, 4599, 163632, 1892670, 9729720, 28161819, 51126740, 61555824, 50308245, 27806065, 10023948, 2143428, 208012];
...
in which column 1 appears to equal A000716, the coefficients in P(x)^3,
and column 2 appears to equal twice A023005, the coefficients in P(x)^6,
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + ... + A000041(n)*x^n + ...
Also, the power series expansions of P(x)^3 and P(x)^6 begin
P(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 + 429*x^7 + 810*x^8 + 1479*x^9 + 2640*x^10 + ... + A000716(n)*x^n + ...
P(x)^6 = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2492*x^6 + 6372*x^7 + 15525*x^8 + 36280*x^9 + 81816*x^10 + ... + A023005(n)*x^n + ...
The main diagonal equals the Catalan numbers (A000108), where g.f. C(x) = 1 + x*C(x)^2 begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..5150

Cf. A000108 (main diagonal), A000041, A000716, A023005.
Cf. A355361 (y=1), A355362 (y=2), A355363 (y=3), A355364, A355365.
Cf. A355350 (related table).

{T(n,k) = my(A=[1,y],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( x*y*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));polcoeff(A[n+1],k,y)}
for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))

G.f. A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*y^k satisfies:
(1) x*y*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) -x*y*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x,y)^n.
(3) x*y*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
(4) A(x,y) = B(x, y*A(x,y)) and A(x, y/B(x,y)) = B(x,y) where x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * B(x,y)^n, and B(x,y) is the g.f. of table A355350.

A000713 EULER transform of 3, 2, 2, 2, 2, 2, 2, 2, ...

Original entry on oeis.org

1, 3, 8, 18, 38, 74, 139, 249, 434, 734, 1215, 1967, 3132, 4902, 7567, 11523, 17345, 25815, 38045, 55535, 80377, 115379, 164389, 232539, 326774, 456286, 633373, 874213, 1200228, 1639418, 2228546, 3015360, 4062065, 5448995, 7280060, 9688718, 12846507, 16972577

Offset: 0

N. J. A. Sloane

nonn

Equals row sums of triangle A146023. - Gary W. Adamson, Oct 26 2008
Partial sums of A000712. - Geoffrey Critzer, Apr 19 2012, corrected by Omar E. Pol, Jun 19 2012
Equals the number of partitions of n with 1's of three kinds and all parts >1 of two kinds. - Gregory L. Simay, Mar 25 2018

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 390
N. J. A. Sloane, Transforms

Cf. A000041, A000716.
Row sums of triangle A093010.
Cf. A146023. - Gary W. Adamson, Oct 26 2008

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<2,3,2)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

nn=20; g=Product[1/(1-x^i), {i,1,nn}]; c=1/(1-x); CoefficientList[Series[g^2/(1-x), {x,0,nn}], x] (* Geoffrey Critzer, Apr 19 2012 *)

x='x+O('x^66); Vec(1/((1-x)*eta(x)^2)) \\ Joerg Arndt, May 01 2013

from functools import lru_cache
from sympy import divisor_sigma
@lru_cache(maxsize=None)
def A000713(n): return sum(A000713(k)*((divisor_sigma(n-k)<<1)+1) for k in range(n))//n if n else 1 # Chai Wah Wu, Sep 25 2023

G.f.: A(x)/(1-x) where A(x) is g.f. for A000712. - Geoffrey Critzer, Apr 19 2012.
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ sqrt(3*n)/Pi * A000712(n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*Pi*3^(1/4)*n^(3/4)).
(End)
G.f.: exp(Sum_{k>=1} (2*sigma_1(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018

Extended with formula from Christian G. Bower, Apr 15 1998

Definition changed by N. J. A. Sloane, Aug 15 2006

A053762 Number of 3-colored generalized Frobenius partitions of n.

Original entry on oeis.org

1, 9, 27, 82, 207, 486, 1055, 2205, 4374, 8427, 15696, 28539, 50630, 88119, 150417, 252727, 418068, 682344, 1099343, 1750968, 2758185, 4301682, 6645150, 10175625, 15451744, 23281686, 34819227, 51712860, 76292784, 111850740, 162997314

Offset: 0

James Sellers, Apr 05 2000

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + 9*x + 27*x^2 + 82*x^3 + 207*x^4 + 486*x^5 + 1055*x^6 + 2205*x^7 + ...
1/q + 9*q^7 + 27*q^15 + 82*q^23 + 207*q^31 + 486*q^39 + 1055*q^47 + 2205*q^55 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. E. Andrews, Generalized Frobenius Partitions, AMS Memoir 301, 1984 (sequence is denoted c\phi_3(n)).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000716, A004016, A051136.

nmax = 30; CoefficientList[Series[(Product[(1 - x^k)^3, {k, 1, nmax}] + 9*x*Product[(1 - x^(9*k))^3, {k, 1, nmax}]) / Product[((1 - x^k)^3*(1 - x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)
a[n_]:= SeriesCoefficient[q^(1/8)*(eta[q]^3 + 9*eta[q^9]^3)/(eta[q]^3* eta[q^3]), {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Feb 08 2018 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / (eta(x + A)^3 * eta(x^3 + A)), n))} /* Michael Somos, Mar 09 2011 */

Expansion of q^(1/8) * (eta(q)^3 + 9 * eta(q^9)^3) / (eta(q)^3 * eta(q^3)) in powers of q. - Michael Somos, Mar 09 2011
Expansion of a(x) / f(-x)^3 in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function. - Michael Somos, Aug 21 2012
Convolution of A000716 and A004016. - Michael Somos, Mar 09 2011
a(n) ~ exp(sqrt(2*n)*Pi)/(4*sqrt(3)*n). - Vaclav Kotesovec, Nov 13 2016

cp's OEIS Frontend

A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

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A000991 Number of 3-line partitions of n.

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A341221 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^3.

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A355350 G.f. A(x,y) satisfies: x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

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A339318 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^3.

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A356500 Coefficients T(n,k) of x^n*y^k in A(x,y) for n >= 0, k = 0..3*n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

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A385001 Irregular triangle read by rows: T(n,k) is the number of partitions of n with k designated summands, n >= 0, 0 <= k <= A003056(n).

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A355350 G.f. A(x,y) satisfies: xy = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

A356500 Coefficients T(n,k) of x^ny^k in A(x,y) for n >= 0, k = 0..3n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

A355360 G.f. A(x,y) satisfies: xyA(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.