A257673
Triangle T(n,k), n>=0, 0<=k<=n, read by rows: row n is the inverse binomial transform of the n-th row of array A255961, which has the Euler transform of (j->j*k) in column k.
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 6, 6, 1, 0, 13, 21, 9, 1, 0, 24, 62, 45, 12, 1, 0, 48, 162, 174, 78, 15, 1, 0, 86, 396, 576, 376, 120, 18, 1, 0, 160, 917, 1719, 1509, 695, 171, 21, 1, 0, 282, 2036, 4761, 5340, 3285, 1158, 231, 24, 1, 0, 500, 4380, 12441, 17234, 13473, 6309, 1792, 300, 27, 1
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 6, 6, 1;
0, 13, 21, 9, 1;
0, 24, 62, 45, 12, 1;
0, 48, 162, 174, 78, 15, 1;
0, 86, 396, 576, 376, 120, 18, 1;
0, 160, 917, 1719, 1509, 695, 171, 21, 1;
0, 282, 2036, 4761, 5340, 3285, 1158, 231, 24, 1;
...
Columns k=0-10 give:
A000007,
A000219 (for n>0),
A321947,
A321948,
A321949,
A321950,
A321951,
A321952,
A321953,
A321954,
A321955.
-
A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
# Uses function PMatrix from A357368.
PMatrix(10, A000219); # Peter Luschny, Oct 19 2022
-
A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 21 2017, translated from Maple *)
A307062
Expansion of 1/(2 - Product_{k>=1} (1 + x^k)^k).
Original entry on oeis.org
1, 1, 3, 10, 29, 88, 264, 790, 2366, 7086, 21216, 63523, 190201, 569485, 1705121, 5105383, 15286247, 45769238, 137039743, 410316854, 1228548190, 3678451550, 11013817655, 32976968175, 98737827756, 295635383297, 885175234817, 2650343093602, 7935511791620, 23760073760720, 71141108467679
Offset: 0
-
m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+x^j)^j: j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024
-
b:= proc(n) b(n):= add((-1)^(n/d+1)*d^2, d=numtheory[divisors](n)) end:
g:= proc(n) g(n):= `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n) end:
a:= proc(n) a(n):= `if`(n=0, 1, add(g(k)*a(n-k), k=1..n)) end:
seq(a(n), n=0..45); # Alois P. Heinz, Jan 24 2024
-
nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]
-
m=80;
def f(x): return 1/( 2 - product((1+x^j)^j for j in range(1,m+3)) )
def A307062_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( f(x) ).list()
A307062_list(m) # G. C. Greubel, Jan 24 2024
A320652
Expansion of 1/(2 - Product_{k>=1} 1/(1 - k*x^k)).
Original entry on oeis.org
1, 1, 4, 13, 45, 147, 497, 1643, 5490, 18252, 60812, 202364, 673915, 2243295, 7468973, 24865272, 82783967, 275605513, 917563193, 3054785032, 10170143277, 33858882922, 112724577088, 375287739083, 1249425198725, 4159643200494, 13848474406054, 46104972636634, 153494780854254
Offset: 0
-
a:=series(1/(2-mul(1/(1-k*x^k),k=1..100)),x=0,29): seq(coeff(a,x,n),n=0..28); # Paolo P. Lava, Apr 02 2019
-
nmax = 28; CoefficientList[Series[1/(2 - Product[1/(1 - k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - Sum[k x^k/Product[(1 - j x^j), {j, 1, k}], {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Total[Times@@@IntegerPartitions[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 28}]
Showing 1-3 of 3 results.
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