A246935
Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 14, 5, 0, 1, 5, 20, 39, 34, 7, 0, 1, 6, 30, 84, 129, 74, 11, 0, 1, 7, 42, 155, 356, 399, 166, 15, 0, 1, 8, 56, 258, 805, 1444, 1245, 350, 22, 0, 1, 9, 72, 399, 1590, 4055, 5876, 3783, 746, 30, 0
Offset: 0
A(2,2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
0, 2, 6, 12, 20, 30, 42, 56, ...
0, 3, 14, 39, 84, 155, 258, 399, ...
0, 5, 34, 129, 356, 805, 1590, 2849, ...
0, 7, 74, 399, 1444, 4055, 9582, 19999, ...
0, 11, 166, 1245, 5876, 20455, 57786, 140441, ...
0, 15, 350, 3783, 23604, 102455, 347010, 983535, ...
Columns k=0-10 give:
A000007,
A000041,
A070933,
A242587,
A246936,
A246937,
A246938,
A246939,
A246940,
A246941,
A246942.
-
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; A[n_, k_] := b[n, n, k]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)
A338673
Expansion of Product_{k>=1} 1 / (1 - 4^(k-1)*x^k).
Original entry on oeis.org
1, 1, 5, 21, 101, 421, 2021, 8421, 39397, 167397, 766437, 3244517, 14881253, 62804453, 283415013, 1210159589, 5401907685, 22966866405, 102497423845, 435085808101, 1925197238757, 8215432696293, 36068400468453, 153579729097189, 674546796630501, 2866238341681637, 12508012102193637
Offset: 0
-
nmax = 26; CoefficientList[Series[Product[1/(1 - 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 4^(n - k), {k, 0, n}], {n, 0, 26}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]
A303391
Expansion of Product_{k>=1} (1 + 4*x^k)/(1 - 4*x^k).
Original entry on oeis.org
1, 8, 40, 200, 872, 3720, 15400, 62920, 254440, 1024648, 4112680, 16483400, 66000360, 264150920, 1056903080, 4228272200, 16914393832, 67660396040, 270647139240, 1082600410440, 4330424811880, 17321748357640, 69287088965800, 277148557003720, 1108594618342760
Offset: 0
-
N:= 50: # for a(0)..a(N)
G:= mul((1+4*x^k)/(1-4*x^k),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Feb 13 2019
-
nmax = 25; CoefficientList[Series[Product[(1+4*x^k)/(1-4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
A265975
Expansion of Product_{k>=1} 1/(1 - 4*k*x^k).
Original entry on oeis.org
1, 4, 24, 108, 512, 2164, 9464, 39004, 163008, 663588, 2713752, 10954764, 44328512, 178160724, 716821752, 2874497660, 11532111232, 46187508676, 185028540696, 740595436652, 2964628293504, 11862432443764, 47467812675320, 189902835709212, 759756868215872
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0 or i=1,
4^n, b(n, i-1) +i*4*b(n-i, min(n-i, i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..32); # Alois P. Heinz, Aug 23 2019
-
nmax=40; CoefficientList[Series[Product[1/(1-4*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
A303392
Expansion of Product_{k>=1} ((1 + 4*x^k) / (1 - 4*x^k))^(1/2).
Original entry on oeis.org
1, 4, 12, 52, 156, 612, 2028, 7892, 27324, 107396, 384844, 1520436, 5566876, 22069796, 81990252, 325707348, 1222582268, 4862950020, 18395472460, 73233825524, 278700724764, 1110232691108, 4245596648876, 16920914168148, 64963831455996, 259012955299396
Offset: 0
-
nmax = 30; CoefficientList[Series[Product[((1+4*x^k)/(1-4*x^k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[(-3*QPochhammer[-4, x] / (5*QPochhammer[4, x]))^(1/2), {x, 0, nmax}], x]
Showing 1-5 of 5 results.
Comments