A000716 -id:A000716 - cp's OEIS frontend

A298311 Expansion of Product_{k>=1} 1/((1 - x^(2k))(1 - x^(2*k-1))^3).

1, 3, 7, 16, 32, 61, 112, 197, 336, 560, 912, 1456, 2287, 3536, 5392, 8123, 12096, 17824, 26016, 37632, 53984, 76848, 108601, 152432, 212592, 294704, 406201, 556864, 759488, 1030784, 1392496, 1872784, 2508048, 3345184, 4444384, 5882747, 7758736, 10197712, 13358944, 17444256, 22708719

Offset: 0

Ilya Gutkovskiy, Jan 17 2018

Number of partitions of n where there are 3 kinds of odd parts.

Convolution of the sequences A000009 and A015128.

Michael D. Hirschhorn and James A. Sellers, A Family of Congruences Modulo 7 for Partitions with Monochromatic Even Parts and Multi-Colored Odd Parts, arXiv:2507.09752 [math.CO], 2025. See p. 2.
Index entries for related partition-counting sequences

Cf. A000009, A000041, A000716, A015128, A029862, A029863, A160461, A182818, A278690.

nmax = 40; CoefficientList[Series[Product[1/((1 - x^(2 k)) (1 - x^(2 k - 1))^3), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Product[(1 + x^k)^2/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/((1 - x^(2*k))*(1 - x^(2*k-1))^3).
G.f.: Product_{k>=1} (1 + x^k)^2/(1 - x^k).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2^(5/2)*sqrt(3)*n). - Vaclav Kotesovec, Apr 08 2018
G.f.: 1/Product_{n > = 1} ( 1 - x^(n/gcd(n,k)) ) for k = 4. Cf. A000041 (k = 1), A015128 (k = 2), A278690 (k = 3) and A160461 (k = 5). - Peter Bala, Nov 17 2020

A060850 Array of the coefficients A(n,k) in the expansion of Product_{i>=1} 1/(1-x^i)^n = Sum_{k>=0} A(n,k)*x^k, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 9, 10, 5, 1, 5, 14, 22, 20, 7, 1, 6, 20, 40, 51, 36, 11, 1, 7, 27, 65, 105, 108, 65, 15, 1, 8, 35, 98, 190, 252, 221, 110, 22, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 1, 11, 65, 255

Offset: 1

Bo T. Ahlander (ahlboa(AT)isk.kth.se), May 03 2001

Table read by antidiagonals: entry (n,k) gives number of partitions of n objects into parts of k kinds. - Franklin T. Adams-Watters, Dec 28 2006

			Table (row k, k >= 0: number of partitions of n, n >= 0, into parts of k kinds):
Array begins:
=======================================================================
k\n| 0   1   2    3     4     5      6       7       8       9       10
---|-------------------------------------------------------------------
1  | 1   1   2    3     5     7     11      15      22      30       42
2  | 1   2   5   10    20    36     65     110     185     300      481
3  | 1   3   9   22    51   108    221     429     810    1479     2640
4  | 1   4  14   40   105   252    574    1240    2580    5180    10108
5  | 1   5  20   65   190   506   1265    2990    6765   14725    31027
6  | 1   6  27   98   315   918   2492    6372   15525   36280    81816
7  | 1   7  35  140   490  1547   4522   12405   32305   80465   192899
8  | 1   8  44  192   726  2464   7704   22528   62337  164560   417140
9  | 1   9  54  255  1035  3753  12483   38709  113265  315445   841842
10 | 1  10  65  330  1430  5512  19415   63570  195910  573430  1605340
11 | 1  11  77  418  1925  7854  29183  100529  325193  997150  2919411
  ...
Triangle (row n, n >= 0: number of partitions of n into parts of n - k kinds, 0 <= k <= n) (antidiagonals of above table) (parenthesized last term on each row, which would correspond to row k = 0 in above table)
Triangle begins: (column k: n - k kinds of parts)
===================================
n\k| 0   1   2   3   4   5   6   7
---+-------------------------------
0  |(1)
1  | 1, (0)
2  | 1,  1, (0)
3  | 1,  2,  2, (0)
4  | 1,  3,  5,  3, (0)
5  | 1,  4,  9, 10,  5, (0)
6  | 1,  5, 14, 22, 20,  7, (0)
7  | 1,  6, 20, 40, 51, 36, 11, (0)
  ...

Cf. A067687 (table antidiagonal sums, triangle row sums).
Rows (table), diagonals (triangle): A000041, A000712, A000716, A023003-A023021, A006922.
Columns (table, triangle): A000012, A001477, A000096, A006503, A006504.

t[n_, k_] := CoefficientList[ Series[ Product[1/(1 - x^i)^n, {i, k}], {x, 0, k}], x][[k]]; (* Robert G. Wilson v, Aug 08 2018 *)
t[n_, k_]; = IntegerPartitions[n, {k}]; Table[ t[n - k + 1, k], {n, 12}, {k, n}] // Flatten (* Robert G. Wilson v, Aug 08 2018 *)

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=1..n} A000041(k-1)*A(n-k;x)*x^(k-1), A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004

More terms from Vladeta Jovovic, Jan 02 2004

A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 18, 13, 5, 1, 1, 19, 38, 35, 19, 6, 1, 1, 30, 74, 86, 59, 26, 7, 1, 1, 45, 139, 194, 164, 91, 34, 8, 1, 1, 67, 249, 415, 416, 281, 132, 43, 9, 1, 1, 97, 434, 844, 990, 787, 447, 183, 53, 10, 1

Offset: 0

Omar E. Pol, Jun 27 2012

It appears that row 2 is A034856.
Observation:
Column 1 is the EULER transform of 2,1,1,1,1,1,1,1...
Column 2 is the EULER transform of 3,2,2,2,2,2,2,2...

			Array begins:
1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
1,   4,   8,  13,  19,  26,  34,  43,  53,
1,   7,  18,  35,  59,  91, 132, 183,
1,  12,  38,  86, 164, 281, 447,
1,  19,  74, 194, 416, 787,
1,  30, 139, 415, 990,
1,  45, 249, 844,
1,  67, 434,
1,  97,
1,

Alois P. Heinz, Rows n = 0..140, flattened

Columns (0-3): A000012, A000070, A000713, A210843.
Rows (0-1): A000012, A000027.
Main diagonal gives A303070.
Cf. A000007, A000041, A005758, A006922, A000712, A000716, A023003-A023021, A144064, A195825, A211970.

with(numtheory):
etr:= proc(p) local b;
        b:= proc(n) option remember; `if`(n=0, 1,
              add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n)
            end
      end:
A:= (n, k)-> etr(j-> k +`if`(j=1, 1, 0))(n):
seq(seq(A(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, May 20 2013

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[{j}, k + If[j == 1, 1, 0]]][n]; Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

A353690 Irregular triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A353689 multiplied by A000330(k), and the first element of column k is in row A000217(k).

Original entry on oeis.org

1, 5, 18, 5, 53, 25, 139, 90, 333, 265, 14, 748, 695, 70, 1592, 1665, 252, 3246, 3740, 742, 6379, 7960, 1946, 30, 12152, 16230, 4662, 150, 22524, 31895, 10472, 540, 40764, 60760, 22288, 1590, 72213, 112620, 45444, 4170, 125505, 203820, 89306, 9990, 55, 214378, 361065, 170128, 22440, 275

Offset: 1

Omar E. Pol, May 04 2022

The alternating sum of the n-th row equals A175254(n), the volume of the stepped pyramid with n levels described in A245092, also the n-th term of the convolution of A000203 and A000027.
Column k is the partial sums of the k-th column of the triangle A249120.
Another triangle with the same row lengths and whose alternating row sums give A175254 is A262612.

			Triangle begins:
        1;
        5;
       18,       5;
       53,      25;
      139,      90;
      333,     265,      14;
      748,     695,      70;
     1592,    1665,     252;
     3246,    3740,     742;
     6379,    7960,    1946,     30;
    12152,   16230,    4662,    150;
    22524,   31895,   10472,    540;
    40764,   60760,   22288,   1590;
    72213,  112620,   45444,   4170;
   125505,  203820,   89306,   9990,    55;
   214378,  361065,  170128,  22440,   275;
   360473,  627525,  315336,  47760,   990;
   597450, 1071890,  570696,  97380,  2915;
   977196, 1802365, 1010982, 191370,  7645;
  1578852, 2987250, 1757070, 364560, 18315;
  2522157, 4885980, 3001292, 675720, 41140, 91;
  ...
For n = 6 we have that A175254(6) is equal to [1] + [1 + 3] + [1 + 3 + 4] + [1 + 3 + 4 + 7] + [1 + 3 + 4 + 7 + 6] + [1 + 3 + 4 + 7 + 6 + 12] = 1 + 4 + 8 + 15 + 21 + 33 = 82. On the other hand the alternating sum of the 6th row of the triangle is 333 - 265 + 14 = 82, equaling A175254(6).

Column 1 is A353689.
Row n has length A003056(n).
Column k starts in row A000217(k).
The first element in column k is A000330(k).
Alternating row sums give A175254.
Cf. A000203, A000716, A196020, A210843, A236104, A237593, A245092, A249120, A252117, A262612.

A175254(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k).

A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2n+1) Sum_{k=0..3n} T(n,k)y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n.

Original entry on oeis.org

1, 0, 3, -3, 1, 0, 9, -18, 21, -15, 6, -1, 0, 22, -56, 116, -182, 196, -140, 64, -17, 2, 0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5, 0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14, 0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42

Offset: 0

Paul D. Hanna, Jul 19 2022

Row sums equal A000108, the Catalan numbers:
Sum_{k=0..3*n} T(n,k) = A000108(n) for n >= 0.
T(n,3*n) = (-1)^(n-1) * A000108(n-1) for n >= 1 (Catalan numbers).
Conjecture: T(n,1) = A000716(n) for n >= 1 (number of partitions of n into parts of 3 kinds).
The generating functions of some related sequences are given as follows.
(1) A(x,x) = Sum_{n>=0} A355351(n)*x^n.
(2) A(x,2*x) = Sum_{n>=0} A355352(n)*x^n.
(3) A(x,3*x) = Sum_{n>=0} A355353(n)*x^n.
(4) A(x,4*x) = Sum_{n>=0} A355354(n)*x^n.
(5) A(x,5*x) = Sum_{n>=0} A355355(n)*x^n.
(6) A(x,x^2) = Sum_{n>=0} A355356(n)*x^n.
(7) A(x^2,x) = Sum_{n>=0} A355357(n)*x^n.
(8) A(x,x*y) = Sum_{n>=0} x^n * Sum_{k=0..n} A355350(n,k) * y^k.
(9) 1/A(4*x,-1) = 2*Sum_{n>=0} A268300(n)*x^n.
(10) A(x,2) = -Sum_{n>=0} A355871(n)*x^n.
SPECIFIC VALUES.
(V.1) A(x,y) = -exp(-Pi) at x = exp(-2*Pi) and y = exp(Pi) * Pi^(1/4)/gamma(3/4).
(V.2) A(x,y) = -exp(-2*Pi) at x = exp(-4*Pi) and y = exp(2*Pi) * Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) A(x,y) = -exp(-3*Pi) at x = exp(-6*Pi) and y = exp(3*Pi) * Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) A(x,y) = -exp(-4*Pi) at x = exp(-8*Pi) and y = exp(4*Pi) * Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) A(x,y) = -exp(-sqrt(3)*Pi) at x = exp(-2*sqrt(3)*Pi) and y = exp(sqrt(3)*Pi) * gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

			G.f.: A(x,y) = 1/(1-y) + x*(y^3 - 3*y^2 + 3*y)/(1-y)^3 + x^2*(-y^6 + 6*y^5 - 15*y^4 + 21*y^3 - 18*y^2 + 9*y)/(1-y)^5 + x^3*(2*y^9 - 17*y^8 + 64*y^7 - 140*y^6 + 196*y^5 - 182*y^4 + 116*y^3 - 56*y^2 + 22*y)/(1-y)^7 + x^4*(-5*y^12 + 57*y^11 - 297*y^10 + 934*y^9 - 1971*y^8 + 2934*y^7 - 3148*y^6 + 2436*y^5 - 1329*y^4 + 496*y^3 - 144*y^2 + 51*y)/(1-y)^9 + x^5*(14*y^15 - 200*y^14 + 1330*y^13 - 5455*y^12 + 15412*y^11 - 31724*y^10 + 49060*y^9 - 57914*y^8 + 52462*y^7 - 36302*y^6 + 18846*y^5 - 7005*y^4 + 1680*y^3 - 270*y^2 + 108*y)/(1-y)^11 + ...
where
y = ... + 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,
y = (1 - x*A(x,y))*(1 - 1/A(x,y))*(1-x) * (1 - x^2*A(x,y))*(1 - x/A(x,y))*(1-x^2) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y))*(1-x^3) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y))*(1-x^4) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y))*(1-x^n) * ...
This triangle of coefficients T(n,k) of x^n*y^k/(1-y)^(2*n+1) in A(x,y), for k = 0..3*n in row n, begins
n = 0: [1];
n = 1: [0, 3, -3, 1];
n = 2: [0, 9, -18, 21, -15, 6, -1];
n = 3: [0, 22, -56, 116, -182, 196, -140, 64, -17, 2];
n = 4: [0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5];
n = 5: [0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14];
n = 6: [0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42];
n = 7: [0, 429, -63, 18281, -134985, 594399, -1941037, 4947447, -10062669, 16571700, -22316250, 24716922, -22564425, 16956135, -10435305, 5210319, -2078910, 647565, -151825, 25215, -2646, 132]; ...
The rightmost border equals the signed Catalan numbers (A000108) shifted right one place.
Column 1 appears to equal A000716 (ignoring the initial term).
Example: at y = x, we have the g.f. of A355351:
A(x,x) = 1/(1-x) + x*(3*x - 3*x^2 + x^3)/(1-x)^3 + x^2*(9*x - 18*x^2 + 21*x^3 - 15*x^4 + 6*x^5 - x^6)/(1-x)^5 + x^3*(22*x - 56*x^2 + 116*x^3 - 182*x^4 + 196*x^5 - 140*x^6 + 64*x^7 - 17*x^8 + 2*x^9)/(1-x)^7 + ... = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + ... + A355351(n)*x^n + ...
where x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,x)^n.

Paul D. Hanna, Table of n, a(n) for n = 0..3875

Cf. A000108 (row sums), A355871 (y=2).
Cf. A355350 (related triangle), A355351 (y=x), A355352 (y=2*x), A355353 (y=3*x), A355354 (y=4*x), A355355 (y=5*x), A355356 (y=x^2), A355357 (x=x^2,y=x).
Cf. A355360 (related triangle), A000716.

{T(n,k) = my(A=[1/(1-y)],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( (y - sum(m=-t,t, (-1)^m * x^(m*(m+1)/2) * Ser(A)^m )), #A-1,x)/(1-y)^2);polcoeff(A[n+1]*(1-y)^(2*n+1),k,y)}
for(n=0,12, for(k=0,3*n, print1( T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) y = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^n), by the Jacobi triple product identity.

A356501 Coefficients T(n,k) of x^(4n+1-k)y^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

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

Offset: 0

Paul D. Hanna, Aug 09 2022

Row sums equal A355872.
Alternating row sums equals zero for all rows.
Rightmost border equals [x^n*y^(3*n+1)] A(x,y) = A002293(n) = binomial(4*n, n)/(3*n + 1).
This triangle may be formed from the nonzero antidiagonals of triangle A356500; see main entry A356500 for further formulas for the coefficients in g.f. A(x,y).

			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 triangle of coefficients of x^(4*n+1-k)*y^k in A(x,y) for n >= 0, k = 0..3*n+1, begins:
n = 0: [1, 1];
n = 1: [0, 3, 6, 4, 1];
n = 2: [0, 9, 54, 120, 135, 84, 28, 4];
n = 3: [0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22];
n = 4: [0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140];
n = 5: [0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969];
n = 6: [0, 221, 14952, 282948, 2578147, 14039250, 51126740, 133101836, 258436719, 384735141, 446971668, 409367712, 296679006, 169713208, 75904032, 26050408, 6640711, 1187433, 133266, 7084];
...

Paul D. Hanna, Table of n, a(n) for n = 0..671

Cf. A355872 (row sums), A356500 (main entry), A002293 (right border), A000716 (column 1).

{T(n,k) = my(A=[y],M); for(i=1,4*n+1, A = concat(A,0); M = ceil(sqrt(4*n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[4*n+2-k],k,y)}
for(n=0,7, 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.

A382025 Triangle read by rows: T(n, k) is the number of partitions of n with at most k parts where 0 <= k <= n, and each part is one of three kinds.

Original entry on oeis.org

1, 0, 3, 0, 3, 9, 0, 3, 12, 22, 0, 3, 18, 36, 51, 0, 3, 21, 57, 87, 108, 0, 3, 27, 82, 148, 193, 221, 0, 3, 30, 111, 225, 330, 393, 429, 0, 3, 36, 144, 333, 528, 681, 765, 810, 0, 3, 39, 184, 460, 808, 1106, 1316, 1424, 1479, 0, 3, 45, 225, 630, 1182, 1740, 2163, 2439, 2574, 2640

Offset: 0

Peter Dolland, Mar 12 2025

The 1-kind case is Euler's table A026820.

The 2-kind case is A381895.

			Triangle starts:
 0 : [1]
 1 : [0, 3]
 2 : [0, 3,  9]
 3 : [0, 3, 12,  22]
 4 : [0, 3, 18,  36,  51]
 5 : [0, 3, 21,  57,  87,  108]
 6 : [0, 3, 27,  82, 148,  193,  221]
 7 : [0, 3, 30, 111, 225,  330,  393,  429]
 8 : [0, 3, 36, 144, 333,  528,  681,  765,  810]
 9 : [0, 3, 39, 184, 460,  808, 1106, 1316, 1424, 1479]
10 : [0, 3, 45, 225, 630, 1182, 1740, 2163, 2439, 2574, 2640]
...

Main diagonal gives A000716.

Cf. A026820, A381895.

from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
kinds = 3 - 1   # the number of part kinds - 1
def a382025_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( kinds + p[k], kinds)
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

A384926 Number of partitions of n with six designated summands.

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 221, 414, 765, 1344, 2310, 3834, 6248, 9894, 15408, 23550, 35394, 52353, 76402, 109959, 156366, 219850, 305796, 421281, 574568, 777234, 1042083, 1387037, 1831362, 2402595, 3128995, 4051797, 5211639, 6668490, 8482089, 10737063, 13516615

Offset: 21

Omar E. Pol, Jul 23 2025

nonn

			21 has only one partition with six designated summands: [6'+ 5'+ 4'+ 3'+ 2'+ 1'], so a(21) = 1.
22 has three partitions with six designated summands: [7'+ 5'+ 4'+ 3'+ 2'+ 1'], [6'+ 5'+ 4'+ 3'+ 2'+ 1'+ 1], [6'+ 5'+ 4'+ 3'+ 2'+ 1 + 1'], so a(22) = 3.

Alois P. Heinz, Table of n, a(n) for n = 21..4000

Column k=6 of A385001.
Columns of A385001 converge to A000716.
Other columns k of A385001 are A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5).

b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*j*x, j=1..n/i))), x, 7)
    end:
a:= n-> coeff(b(n$2), x, 6):
seq(a(n), n=21..57);  # Alois P. Heinz, Jul 23 2025

nmax=60; Drop[CoefficientList[Series[1/13 * Sum[(-1)^k*(2*k + 1)*Binomial[k + 6, 12]*x^(k*(k + 1)/2), {k, 6, nmax}]/Sum[(-1)^k*(2*k + 1)*x^(k*(k + 1)/2), {k, 0, nmax}], {x, 0, nmax}], x] , 21] (* Vaclav Kotesovec, Jul 29 2025 *)

A000716(n) >= a(21+n) with equality only for n <= 6.

Sum_{k=1..n} a(k) ~ Pi^12 * n^12 / (12! * 13!). - Vaclav Kotesovec, Aug 01 2025

More terms from Alois P. Heinz, Jul 23 2025

A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 8, 1, 1, 15, 4, 1, 21, 13, 1, 33, 28, 1, 1, 41, 58, 4, 1, 56, 103, 13, 1, 69, 170, 35, 1, 87, 269, 77, 1, 1, 99, 404, 158, 4, 1, 127, 579, 298, 13, 1, 141, 810, 529, 35, 1, 165, 1116, 880, 86, 1, 189, 1470, 1431, 183, 1, 1, 220, 1935, 2214, 371, 4, 1, 238, 2475, 3348, 701, 13

Offset: 0

Omar E. Pol, Jul 22 2025

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).
Column k gives the partial sums of the column k of A385001.
Columns converge to A210843 which is also the partial sums of A000716.

			Triangle begins:
---------------------------------------------
   n\k:   0    1     2      3     4    5   6
---------------------------------------------
   0 |    1;
   1 |    1,   1;
   2 |    1,   4;
   3 |    1,   8,    1;
   4 |    1,  15,    4;
   5 |    1,  21,   13;
   6 |    1,  33,   28,     1;
   7 |    1,  41,   58,     4;
   8 |    1,  56,  103,    13;
   9 |    1,  69,  170,    35;
  10 |    1,  87,  269,    77,    1;
  11 |    1,  99,  404,   158,    4;
  12 |    1, 127,  579,   298,   13;
  13 |    1, 141,  810,   529,   35;
  14 |    1, 165, 1116,   880,   86;
  15 |    1, 189, 1470,  1431,  183,   1;
  16 |    1, 220, 1935,  2214,  371,   4;
  17 |    1, 238, 2475,  3348,  701,  13;
  18 |    1, 277, 3156,  4894, 1269,  35;
  19 |    1, 297, 3921,  7036, 2187,  86;
  20 |    1, 339, 4866,  9871, 3639, 194;
  21 |    1, 371, 5906, 13629, 5872, 402,  1;
  ...

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

Columns: A000012 (k=0), A024916 (k=1).

Cf. A000217, A000716, A002024, A003056, A077285, A195825, A210843, A211970, A385001.

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:
g:= proc(n) option remember; `if`(n<0, 0, g(n-1)+b(n$2)) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 22 2025

A382343 Triangle read by rows: T(n, k) is the number of partitions of n into k parts where 0 <= k <= n, and each part is one of 3 kinds.

Original entry on oeis.org

1, 0, 3, 0, 3, 6, 0, 3, 9, 10, 0, 3, 15, 18, 15, 0, 3, 18, 36, 30, 21, 0, 3, 24, 55, 66, 45, 28, 0, 3, 27, 81, 114, 105, 63, 36, 0, 3, 33, 108, 189, 195, 153, 84, 45, 0, 3, 36, 145, 276, 348, 298, 210, 108, 55, 0, 3, 42, 180, 405, 552, 558, 423, 276, 135, 66

Offset: 0

Peter Dolland, Mar 27 2025

			Triangle starts:
 0 : [1]
 1 : [0, 3]
 2 : [0, 3,  6]
 3 : [0, 3,  9,  10]
 4 : [0, 3, 15,  18,  15]
 5 : [0, 3, 18,  36,  30,  21]
 6 : [0, 3, 24,  55,  66,  45,  28]
 7 : [0, 3, 27,  81, 114, 105,  63,  36]
 8 : [0, 3, 33, 108, 189, 195, 153,  84,  45]
 9 : [0, 3, 36, 145, 276, 348, 298, 210, 108,  55]
10 : [0, 3, 42, 180, 405, 552, 558, 423, 276, 135, 66]
...

Main diagonal gives A000217(n+1).
Row sums give A000716.
Cf. A008284 (1-kind), A382342 (2-kind).
Cf. A022598, A382025.

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(x^j*b(n-i*j, min(n-i*j, i-1))*(j+2)*(j+1)/2, j=0..n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 27 2025

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^j*b[n-i*j, Min[n-i*j, i-1]]*(j+2)*(j+1)/2, {j, 0, n/i}]]]];
T[n_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 30 2025, after Alois P. Heinz *)

from sympy import binomial
from sympy.utilities.iterables import partitions
kinds = 3 - 1   # the number of part kinds - 1
def t_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( kinds + p[k], kinds)
        if s > 0 :
            t[s - 1] += fact
    return [0] + t

T(n,n) = binomial(n + 2, 2) = A000217(n + 1).
T(n,1) = 3 for n >= 1.
T(n,k) = A382025(n,k) - A382025(n,k-1) for 1 <= k <= n.
Sum_{k=0..n} (-1)^k * T(n,k) = A022598(n). - Alois P. Heinz, Mar 27 2025

cp's OEIS Frontend

A298311 Expansion of Product_{k>=1} 1/((1 - x^(2*k))*(1 - x^(2*k-1))^3).

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A060850 Array of the coefficients A(n,k) in the expansion of Product_{i>=1} 1/(1-x^i)^n = Sum_{k>=0} A(n,k)*x^k, n >= 1, k >= 0.

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A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

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A353690 Irregular triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A353689 multiplied by A000330(k), and the first element of column k is in row A000217(k).

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A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.

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A356501 Coefficients T(n,k) of x^(4*n+1-k)*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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A382025 Triangle read by rows: T(n, k) is the number of partitions of n with at most k parts where 0 <= k <= n, and each part is one of three kinds.

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A384926 Number of partitions of n with six designated summands.

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A298311 Expansion of Product_{k>=1} 1/((1 - x^(2k))(1 - x^(2*k-1))^3).

A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2n+1) Sum_{k=0..3n} T(n,k)y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n.

A356501 Coefficients T(n,k) of x^(4n+1-k)y^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.