A023005 -id:A023005 - cp's OEIS frontend

A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 5, 0, 1, 5, 14, 22, 20, 7, 0, 1, 6, 20, 40, 51, 36, 11, 0, 1, 7, 27, 65, 105, 108, 65, 15, 0, 1, 8, 35, 98, 190, 252, 221, 110, 22, 0, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 0, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 0

Offset: 0

Alois P. Heinz, Sep 09 2008

A(n,k) is also the number of partitions of n into parts of k kinds.
In general, column k > 0 is asymptotic to k^((k+1)/4) * exp(Pi*sqrt(2*k*n/3)) / (2^((3*k+5)/4) * 3^((k+1)/4) * n^((k+3)/4)) * (1 - (Pi*k^(3/2)/(24*sqrt(6)) + sqrt(3)*(k+1)*(k+3)/(8*Pi*sqrt(2*k))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field with k elements that contain an upper-triangular matrix. - Geoffrey Critzer, Nov 11 2022

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5, ...
  0,   2,   5,   9,  14,  20, ...
  0,   3,  10,  22,  40,  65, ...
  0,   5,  20,  51, 105, 190, ...
  0,   7,  36, 108, 252, 506, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48. See Table 1.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-24 give: A000007, A000041, A000712, A000716, A023003, A023004, A023005, A023006, A023007, A023008, A023009, A023010, A005758, A023011, A023012, A023013, A023014, A023015, A023016, A023017, A023018, A023019, A023020, A023021, A006922.
Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).
Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.
Main diagonal gives A008485.
Antidiagonal sums give A067687.
Cf. A060642, A122768, A246935, A255961, A261718.

# DedekindEta is defined in A000594.
A144064Column(k, len) = DedekindEta(len, -k)
for n in 0:8 A144064Column(n, 6) |> println end # Peter Luschny, Mar 10 2018

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)(n): seq(seq(A(n, d-n), n=0..d), d=0..14);

a[0, ] = 1; a[, 0] = 0; a[n_, k_] := SeriesCoefficient[ Product[1/(1 - x^j)^k, {j, 1, n}], {x, 0, n}]; Table[a[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 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[k&][n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

Mat(apply( {A144064_col(k,nMax=9)=Col(1/eta('x+O('x^nMax))^k,nMax)}, [0..9])) \\ M. F. Hasler, Aug 04 2024

G.f. of column k: Product_{j>=1} 1/(1-x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i):

A341225 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^6.

Original entry on oeis.org

1, 12, 78, 370, 1437, 4848, 14719, 41148, 107610, 266296, 628941, 1427118, 3127369, 6646440, 13746081, 27744926, 54782271, 106029918, 201512970, 376630680, 693161334, 1257641676, 2251764699, 3982196910, 6961522279, 12038699766, 20607718317, 34938910360

Offset: 6

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..10000

Column k=6 of A060642.

Cf. A000041, A023005, A048574, A327384, A341221, A341222, A341223, 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, 6):
seq(a(n), n=6..33);  # Alois P. Heinz, Feb 07 2021

nmax = 33; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

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.

A339321 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^6.

Original entry on oeis.org

1, 6, 6, 27, 6, 42, 6, 98, 27, 42, 6, 204, 6, 42, 42, 315, 6, 204, 6, 204, 42, 42, 6, 792, 27, 42, 98, 204, 6, 330, 6, 918, 42, 42, 42, 1044, 6, 42, 42, 792, 6, 330, 6, 204, 204, 42, 6, 2682, 27, 204, 42, 204, 6, 792, 42, 792, 42, 42, 6, 1716, 6, 42, 204, 2492, 42, 330

Offset: 1

Ilya Gutkovskiy, Nov 30 2020

nonn

Number of factorizations of n into factors (greater than 1) of 6 kinds.

Cf. A001055, A023005, A301830, A339318, A339319, A339320, A339322, A339323, A339324.

a(p^k) = A023005(k) for prime p.

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.

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A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

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A341225 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^6.

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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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A339321 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^6.

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A355360 G.f. A(x,y) satisfies: x*y*A(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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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.

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.