A024429 -id:A024429 - cp's OEIS frontend

A121867 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)A(k); entry gives A sequence (cf. A121868).

1, 0, -1, -3, -6, -5, 33, 266, 1309, 4905, 11516, -22935, -556875, -4932512, -32889885, -174282151, -612400262, 907955295, 45283256165, 573855673458, 5397236838345, 41604258561397, 250231901787780, 756793798761989, -8425656230853383, -213091420659985440, -2990113204010882473

Offset: 0

N. J. A. Sloane, Sep 05 2006

Stirling transform of A056594.

			From _Peter Bala_, Aug 28 2008: (Start)
E_2(k) as linear combination of E_2(i), i = 0..1.
============================
..E_2(k)..|...E_2(0)..E_2(1)
============================
..E_2(2)..|....-1.......1...
..E_2(3)..|....-3.......0...
..E_2(4)..|....-6......-5...
..E_2(5)..|....-5.....-23...
..E_2(6)..|....33.....-74...
..E_2(7)..|...266....-161...
..E_2(8)..|..1309......57...
..E_2(9)..|..4905....3466...
...
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..220
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.

Cf. A121868, A024430, A024429, A056594.

Cf. A000587, A143623, A143624, A143628, A143631. - Peter Bala, Aug 28 2008

List([0..30], n-> Sum([0..Int(n/2)], k-> (-1)^k*Stirling2(n,2*k)) ); # G. C. Greubel, Oct 09 2019

[(&+[(-1)^k*StirlingSecond(n,2*k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 09 2019

# Maple code for A024430, A024429, A121867, A121868.
M:=30; a:=array(0..100); b:=array(0..100); c:=array(0..100); d:=array(0..100); a[0]:=1; b[0]:=0; c[0]:=1; d[0]:=0;
for n from 1 to M do a[n]:=add(binomial(n-1,k)*b[k], k=0..n-1); b[n]:=add(binomial(n-1,k)*a[k], k=0..n-1); c[n]:=add(binomial(n-1,k)*d[k], k=0..n-1); d[n]:=-add(binomial(n-1,k)*c[k], k=0..n-1); od: ta:=[seq(a[n],n=0..M)]; tb:=[seq(b[n],n=0..M)]; tc:=[seq(c[n],n=0..M)]; td:=[seq(d[n],n=0..M)];
# Code based on Stirling transform:
stirtr:= proc(p) proc(n) option remember;
            add(p(k) *Stirling2(n,k), k=0..n) end
         end:
a:= stirtr(n-> (I^n + (-I)^n)/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 29 2011

a[n_] := (BellB[n, -I] + BellB[n, I])/2; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Mar 06 2013, after Alois P. Heinz *)

a(n) = sum(k=0,n\2, (-1)^k*stirling(n,2*k,2));
vector(30, n, a(n-1)) \\ G. C. Greubel, Oct 09 2019

[sum((-1)^k*stirling_number2(n,2*k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 09 2019

From Peter Bala, Aug 28 2008: (Start)
This sequence and its companion A121868 are related to the pair of constants cos(1) + sin(1) and cos(1) - sin(1) and may be viewed as generalizations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587. Define E_2(k) = Sum_{n >= 0} (-1)^floor(n/2) * n^k/n! for k = 0,1,2,... . Then E_2(0) = cos(1) + sin(1) and E_2(1) = cos(1) - sin(1). It is easy to see that E_2(k+2) = E_2(k+1) - Sum_{i = 0..k} 2^i*binomial(k,i)*E_2(k-i) for k >= 0. Hence E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = - E_2(0) + E_2(1), E_2(3) = -3*E_2(0) and E_2(4) = - 6*E_2(0) - 5*E_2(1). More examples are given below.
To find the precise result, show F(k) := Sum_{n >= 0} (-1)^floor((n+1)/2)*n^k/n! satisfies the above recurrence with F(0) = E_2(1) and F(1) = -E_2(0) and then use the identity Sum_{i = 0..k} binomial(k,i)*E_2(i) = -F(k+1) to obtain E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1). For similar results see A143628. The decimal expansions of E_2(0) and E_2(1) are given in A143623 and A143624 respectively. (End)
E.g.f.: A(x) = cos(exp(x)-1).
a(n) = Sum_{k=0..floor(n/2)} stirling2(n,2*k)*(-1)^(k). - Vladimir Kruchinin, Jan 29 2011

A121868 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)A(k); entry gives B sequence (cf. A121867).

Original entry on oeis.org

0, -1, -1, 0, 5, 23, 74, 161, -57, -3466, -27361, -155397, -687688, -1888525, 4974059, 134695952, 1400820897, 11055147275, 70658948426, 327448854237, 223871274083, -19116044475298, -314203665206509, -3562429698724513, -33024521386113840, -250403183401213513

Offset: 0

N. J. A. Sloane, Sep 05 2006

sign

Stirling transform of (I^(n+1)+(-I)^(n+1))/2 = (0,-1,0,1,..) repeated.

			From _Peter Bala_, Aug 28 2008: (Start)
E_2(k) as a linear combination of E_2(i), i = 0..1.
============================
..E_2(k)..|...E_2(0)..E_2(1)
============================
..E_2(2)..|....-1.......1...
..E_2(3)..|....-3.......0...
..E_2(4)..|....-6......-5...
..E_2(5)..|....-5.....-23...
..E_2(6)..|....33.....-74...
..E_2(7)..|...266....-161...
..E_2(8)..|..1309......57...
..E_2(9)..|..4905....3466...
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..250
A. Fekete and G. Martin, Problem 10791: Squared Series Yielding Integers, Amer. Math. Monthly, 108 (No. 2, 2001), 177-178.
V. V. Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A121867, A024430, A024429.

Cf. A000587, A143623, A143624, A143628, A143631.

List([0..30], n-> Sum([0..Int(n/2)], k-> (-1)^(k+1)* Stirling2(n,2*k+1)) ); # G. C. Greubel, Oct 09 2019

[(&+[(-1)^(k+1)*StirlingSecond(n,2*k+1): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Oct 09 2019

# Maple code for A024430, A024429, A121867, A121868.
M:=30; a:=array(0..100); b:=array(0..100); c:=array(0..100); d:=array(0..100); a[0]:=1; b[0]:=0; c[0]:=1; d[0]:=0;
for n from 1 to M do a[n]:=add(binomial(n-1,k)*b[k], k=0..n-1); b[n]:=add(binomial(n-1,k)*a[k], k=0..n-1); c[n]:=add(binomial(n-1,k)*d[k], k=0..n-1); d[n]:=-add(binomial(n-1,k)*c[k], k=0..n-1); od: ta:=[seq(a[n],n=0..M)]; tb:=[seq(b[n],n=0..M)]; tc:=[seq(c[n],n=0..M)]; td:=[seq(d[n],n=0..M)];
# Code based on Stirling transform:
stirtr:= proc(p) proc(n) option remember;
            add(p(k) *Stirling2(n, k), k=0..n) end
         end:
a:= stirtr(n-> (I^(n+1) + (-I)^(n+1))/2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 29 2011

stirtr[p_] := Module[{f}, f[n_] := f[n] = Sum[p[k]*StirlingS2[n, k], {k, 0, n}]; f]; a = stirtr[(I^(#+1)+(-I)^(#+1))/2&]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)
Table[Im[BellB[n, -I]], {n, 0, 25}] (* Vladimir Reshetnikov, Oct 22 2015 *)

a(n) = sum(k=0,n\2, (-1)^(k+1)*stirling(n,2*k+1,2));
vector(30, n, a(n-1)) \\ G. C. Greubel, Oct 09 2019

[sum((-1)^(k+1)*stirling_number2(n,2*k+1) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 09 2019

From Peter Bala, Aug 28 2008: (Start)
This sequence and its companion A121867 are related to the pair of constants cos(1) + sin(1) and cos(1) - sin(1) and may be viewed as generalizations of the Uppuluri-Carpenter numbers (complementary Bell numbers) A000587.
Define E_2(k) = Sum_{n >= 0} (-1)^floor(n/2) *n^k/n! for k = 0,1,2,... . Then E_2(0) = cos(1) + sin(1) and E_2(1) = cos(1) - sin(1). Furthermore, E_2(k) is an integral linear combination of E_2(0) and E_2(1) (a Dobinski-type relation). For example, E_2(2) = - E_2(0) + E_2(1), E_2(3) = -3*E_2(0) and E_2(4) = - 6*E_2(0) - 5*E_2(1). More examples are given below. The precise result is E_2(k) = A121867(k) * E_2(0) - A121868(k) * E_2(1).
For similar results see A143628. The decimal expansions of E_2(0) and E_2(1) are given in A143623 and A143624 respectively. (End)
From Vladimir Kruchinin, Jan 26 2011: (Start)
E.g.f.: A(x) = -sin(exp(x)-1).
a(n) = Sum_{k = 0..floor(n/2)} Stirling2(n,2*k+1)*(-1)^(k+1). (End)

A143817 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)C(k). This entry gives the sequence C(n).

Original entry on oeis.org

0, 0, 1, 3, 7, 16, 46, 203, 1178, 7242, 43786, 259634, 1540540, 9414639, 61061613, 428890726, 3266930298, 26581123093, 226393705465, 1986997358251, 17827284972818, 163278469610570, 1531115974317975, 14771302315885372, 147267150734530892, 1521022490460243316

Offset: 0

Peter Bala, Sep 03 2008

Compare with A024429 and A024430.

This sequence and its companion sequences A(n) = A143815 and B(n) = A143816 may be viewed as generalizations of the Bell numbers A000110. Define R(n) = Sum_{k >= 0} (3k)^n/(3k)! for n = 0,1,2,.... Then the real number R(n) is an integral linear combination of R(0) = 1 + 1/3! + 1/6! + ...., R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... . Some examples are given below. The precise result is R(n) = A(n)*R(0) + B(n)*R(1) + C(n)*(R(2)-R(1)). This generalizes the Dobinski relation for the Bell numbers: Sum_{k >= 0} k^n/k! = A000110(n)*exp(1). See A143815 for more details. Compare with A143628 through A143631. The decimal expansions of R(0), R(2) - R(1) and R(1) may be found in A143819, A143820 and A143821 respectively.

			R(n) as a linear combination of R(0),R(1)
and R(2) - R(1).
=======================================
..R(n)..|.....R(0).....R(1)...R(2)-R(1)
=======================================
..R(3)..|.......1........1........3....
..R(4)..|.......6........2........7....
..R(5)..|......25.......11.......16....
..R(6)..|......91.......66.......46....
..R(7)..|.....322......352......203....
..R(8)..|....1232.....1730.....1178....
..R(9)..|....5672.....8233.....7242....
..R(10).|...32202....39987....43786....

Alois P. Heinz, Table of n, a(n) for n = 0..576
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143816, A143818, A143819, A143820, A143821.

# (1)
M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
a[0]:=1: b[0]:=0: c[0]:=0:
for n from 1 to M do
b[n]:=add(binomial(n-1,k)*a[k], k=0..n-1);
c[n]:=add(binomial(n-1,k)*b[k], k=0..n-1);
a[n]:=add(binomial(n-1,k)*c[k], k=0..n-1);
end do:
A143817:=[seq(c[n], n=0..M)];
# (2)
seq(add(Stirling2(n,3*i+2),i = 0..floor((n-2)/3)), n = 0..24);
# third Maple program:
b:= proc(n, t) option remember; `if`(n=0, irem(t, 2),
      add(b(n-j, irem(t+1, 3))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 20 2018

a[n_] := Sum[ StirlingS2[n, 3*i+2], {i, 0, (n-2)/3}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 06 2013 *)

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, 1)+w*Bell_poly(n, w)+w^2*Bell_poly(n, w^2))/3; \\ Seiichi Manyama, Oct 13 2022

a(n) = Sum_{k = 0..floor((n-2)/3)} Stirling2(n,3k+2).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(exp(x)-1).
A143815(n) + A143816(n) + A143817(n) = Bell(n).
a(n) = ( Bell_n(1) + w * Bell_n(w) + w^2 * Bell_n(w^2) )/3, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/3). - Seiichi Manyama, Oct 13 2022

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A143816 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)C(k). This entry gives the sequence B(n).

Original entry on oeis.org

0, 1, 1, 1, 2, 11, 66, 352, 1730, 8233, 39987, 209793, 1240603, 8287281, 60473869, 463764484, 3647602117, 29165686541, 237499318823, 1984374301872, 17167462137733, 154885317758354, 1461156867801556, 14381004640256202, 146852743814531169, 1546054541191452967

Offset: 0

Peter Bala, Sep 03 2008

Compare with A024429 and A024430.

This sequence and its companion sequences A(n) = A143815 and C(n) = A143817 may be viewed as generalizations of the Bell numbers A000110. Define R(n) = Sum_{k >= 0} (3k)^n/(3k)! for n = 0,1,2,.... Then the real number R(n) is an integral linear combination of R(0) = 1 + 1/3! + 1/6! + ...., R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... and R(1) = 1/2! + 1/5! + 1/8! + .... Some examples are given below. The precise result is R(n) = A(n)*R(0) + B(n)*R(1) + C(n)*(R(2)-R(1)). This generalizes the Dobinski relation for the Bell numbers: Sum_{k >= 0} k^n/k! = A000110(n)*exp(1). See A143815 for more details. Compare with A143628 through A143631. The decimal expansions of R(0), R(2) - R(1) and R(1) may be found in A143819, A143820 and A143821 respectively.

			R(n) as a linear combination of R(0),R(1)
and R(2) - R(1).
=======================================
..R(n)..|.....R(0).....R(1)...R(2)-R(1)
=======================================
..R(3)..|.......1........1........3....
..R(4)..|.......6........2........7....
..R(5)..|......25.......11.......16....
..R(6)..|......91.......66.......46....
..R(7)..|.....322......352......203....
..R(8)..|....1232.....1730.....1178....
..R(9)..|....5672.....8233.....7242....
..R(10).|...32202....39987....43786....

Alois P. Heinz, Table of n, a(n) for n = 0..576
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143817, A143818, A143819, A143820, A143821.

# (1)
M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
a[0]:=1: b[0]:=0: c[0]:=0:
for n from 1 to M do
b[n]:=add(binomial(n-1,k)*a[k], k=0..n-1);
c[n]:=add(binomial(n-1,k)*b[k], k=0..n-1);
a[n]:=add(binomial(n-1,k)*c[k], k=0..n-1);
end do:
A143816:=[seq(b[n], n=0..M)];
# (2)
seq(add(Stirling2(n,3*i+1),i = 0..floor((n-1)/3)), n = 0..24);
# third Maple program:
b:= proc(n, t) option remember; `if`(n=0, irem(t, 2),
      add(b(n-j, irem(t+1, 3))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 20 2018

m = 23; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, b[n] = Sum[Binomial[n - 1, k]*a[k], {k, 0, n - 1}]; c[n] = Sum[Binomial[n - 1, k]*b[k], {k, 0, n - 1}]; a[n] = Sum[Binomial[n - 1, k]*c[k], {k, 0, n - 1}]]; A143816 = Table[ b[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, 1)+w^2*Bell_poly(n, w)+w*Bell_poly(n, w^2))/3; \\ Seiichi Manyama, Oct 13 2022

a(n) = Sum_{k = 0..floor((n-1)/3)} Stirling2(n,3k+1).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(exp(x)-1). A143815(n) + A143816(n) + A143817(n) = Bell(n).
a(n) = ( Bell_n(1) + w^2 * Bell_n(w) + w * Bell_n(w^2) )/3, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/3). - Seiichi Manyama, Oct 13 2022

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A143818 Let R(n) = sum {k = 0..inf} (3k)^n/(3k)! for n = 0,1,2,... . Then the real number R(n) is an integral linear combination of R(0), R(1) and R(2). This sequence gives the coefficients of R(1).

Original entry on oeis.org

0, 1, 0, -2, -5, -5, 20, 149, 552, 991, -3799, -49841, -299937, -1127358, -587744, 34873758, 380671819, 2584563448, 11105613358, -2623056379, -659822835085, -8393151852216, -69959106516419, -390297675629170, -414406919999723

Offset: 1

Peter Bala, Sep 03 2008

The coefficients of R(0) and R(2) are listed in A143815 and A143817 respectively.

			R(n) as a linear combination of R(i),
i = 0..2.
====================================
..R(n)..|.....R(0)....R(1)....R(2)..
====================================
..R(3)..|.......1......-2.......3...
..R(4)..|.......6......-5.......7...
..R(5)..|......25......-5......16...
..R(6)..|......91......20......46...
..R(7)..|.....322.....149.....203...
..R(8)..|....1232.....552....1178...
..R(9)..|....5672.....991....7242...
..R(10).|...32202...-3799...43786...
...

A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143816, A143817.

M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
a[0]:=1: b[0]:=0: c[0]:=0:
for n from 1 to M do
a[n]:=add(binomial(n-1,k)*b[k], k=0..n-1);
b[n]:=add(binomial(n-1,k)*c[k], k=0..n-1);
c[n]:=add(binomial(n-1,k)*a[k], k=0..n-1);
end do:
A143818:=[seq(b[n]-c[n], n=0..M)];

m = 24; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, a[n] = Sum[Binomial[n - 1, k]*b[k], {k, 0, n - 1}]; b[n] = Sum[Binomial[n - 1, k]*c[k], {k, 0, n - 1}]; c[n] = Sum[Binomial[n - 1, k]*a[k], {k, 0, n - 1}] ]; A143818 = Table[c[n] - b[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)

a(n) = A143816(n) - A143817(n). a(n) = sum {k = 0..floor((n-1)/3)} (Stirling2(n,3k+1) - Stirling2(n,3k+2)). Let R(n) = sum {k = 0..inf} (3k)^n/(3k)! for n = 0,1,2,... . Then R(n) = A143815(n)*R(0) + A143818(n)*R(1) + A143817(n)*R(2). Some examples are given below. This generalizes the Dobinski relation for the Bell numbers: sum {k = 0..inf} k^n/k! = A000110(n)*exp(1). See A143815 for more details. Compare with A024429, A024430 and A143628--A143631

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A358824 Number of twice-partitions of n of odd length.

Original entry on oeis.org

0, 1, 2, 4, 7, 15, 32, 61, 121, 260, 498, 967, 1890, 3603, 6839, 12972, 23883, 44636, 82705, 150904, 275635, 501737, 905498, 1628293, 2922580, 5224991, 9296414, 16482995, 29125140, 51287098, 90171414, 157704275, 275419984, 479683837, 833154673, 1442550486, 2493570655

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)        (4)         (5)
       (11)  (21)       (22)        (32)
             (111)      (31)        (41)
             (1)(1)(1)  (211)       (221)
                        (1111)      (311)
                        (2)(1)(1)   (2111)
                        (11)(1)(1)  (11111)
                                    (2)(2)(1)
                                    (3)(1)(1)
                                    (11)(2)(1)
                                    (2)(11)(1)
                                    (21)(1)(1)
                                    (11)(11)(1)
                                    (111)(1)(1)
                                    (1)(1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for set partitions is A024429.
For odd lengths (instead of length) we have A358334.
The case of odd parts also is A358823.
The case of odd sums also is A358826.
The case of odd lengths also is A358834.
For multiset partitions of integer partitions: A358837, ranked by A026424.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A072233, A270995, A271619, A279374, A279785, A306319, A336342, A356932.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&]],{n,0,10}]

R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=vector(n,k,numbpart(k))); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022

G.f.: ((1/Product_{k>=1} (1-A000041(k)*x^k)) - (1/Product_{k>=1} (1+A000041(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 30 2022

A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.

Original entry on oeis.org

0, 1, 1, 7, 37, 271, 2341, 23647, 272917, 3543631, 51123781, 811316287, 14045783797, 263429174191, 5320671485221, 115141595488927, 2657827340990677, 65185383514567951, 1692767331628422661, 46400793659664205567, 1338843898122192101557

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004

Stirling transform of A005212(n)=[1,0,6,0,120,0,5040,...] is a(n)=[1,1,7,37,271,...]. - Michael Somos, Mar 04 2004

Occurs also as first column of a matrix-inversion occurring in a sum-of-like-powers problem. Consider the problem for any fixed natural number m>2 of finding solutions to sum(k=1,n,k^m) = (k+1)^m. Erdos conjectured that there are no solutions for n,m>2. Let D be the matrix of differences of D[m,n] := sum(k=1,n,k^m) - (k+1)^m. Then the generating functions for the rows of this matrix D constitute a set of polynomials in n (for varying n along columns) and the m-th polynomial defining the m-th row. Let GF_D be the matrix of the coefficients of this set of polynomials. Then the present sequence is the (unsigned) second column of GF_D^-1. - Gottfried Helms, Apr 01 2007

			From _Gus Wiseman_, Jan 06 2021: (Start)
a(n) is the number of ordered set partitions of {1..n} into an odd number of blocks. The a(1) = 1 through a(3) = 7 ordered set partitions are:
  {{1}}  {{1,2}}  {{1,2,3}}
                  {{1},{2},{3}}
                  {{1},{3},{2}}
                  {{2},{1},{3}}
                  {{2},{3},{1}}
                  {{3},{1},{2}}
                  {{3},{2},{1}}
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J.-C. Aval, V. Féray, J.-C. Novelli, J.-Y. Thibon, Quasi-symmetric functions as polynomial functions on Young diagrams, arXiv preprint arXiv:1312.2727, 2013
Gottfried Helms, Discussion of a problem concerning summing of like powers

Ordered set partitions are counted by A000670.
The case of (unordered) set partitions is A024429.
The complement (even-length ordered set partitions) is counted by A052841.
A058695 counts partitions of odd numbers, ranked by A300063.
A101707 counts partitions of odd positive rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340102 counts odd-length factorizations into odd factors.
A340692 counts partitions of odd rank.
Other cases of odd length:
- A027193 counts partitions of odd length.
- A067659 counts strict partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
Cf. A000700, A026424, A027187, A028260, A078408, A174725, A236914, A340101.

h := n -> add(combinat:-eulerian1(n,k)*2^k,k=0..n):
a := n -> (h(n)-(-1)^n)/2: seq(a(n),n=0..20); # Peter Luschny, Jul 09 2015

Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]

a(n)=if(n<0,0,n!*polcoeff(subst(y/(1-y^2),y,exp(x+x*O(x^n))-1),n))

{a(n)=polcoeff(sum(m=0,n,(2*m+1)!*x^(2*m+1)/prod(k=1,2*m+1,1-k*x+x*O(x^n))),n)} /* Paul D. Hanna, Jul 20 2011 */

def A089677_list(len):  # with a(0)=1
    e, r = [1], [1]
    for i in (1..len-1):
        for k in range(i-1, -1, -1): e[k] = (e[k]*i)//(i-k)
        r.append(-sum(e[j]*(-1)^(i-j) for j in (0..i-1)))
        e.append(sum(e))
    return r
A089677_list(21) # Peter Luschny, Jul 09 2015

E.g.f.: (exp(x)-1)/(exp(x)*(2-exp(x))).
O.g.f.: Sum_{n>=0} (2*n+1)! * x^(2*n+1) / Product_{k=1..2*n+1} (1-k*x). - Paul D. Hanna, Jul 20 2011
a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n).
a(n) = (A000670(n)-(-1)^n)/2. - Vladeta Jovovic, Jan 17 2005
a(n) ~ n! / (4*(log(2))^(n+1)). - Vaclav Kotesovec, Feb 25 2014
a(n) = Sum_{k=0..floor(n/2)} (2*k+1)!*Stirling2(n, 2*k+1). - Peter Luschny, Sep 20 2015

A340831 Number of factorizations of n into factors > 1 with odd greatest factor.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 2, 3, 1, 1, 2, 2, 1, 3, 1, 2, 4, 1, 1, 1, 2, 2, 2, 2, 1, 4, 2, 2, 2, 1, 1, 4, 1, 1, 4, 0, 2, 3, 1, 2, 2, 2, 1, 4, 1, 1, 4, 2, 2, 3, 1, 3, 5, 1, 1, 5, 2, 1, 2, 3, 1, 5, 2, 2, 2, 1, 2, 1, 1, 2, 4, 4, 1, 3, 1, 3, 5, 1, 1, 6

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

			The a(n) factorizations for n = 45, 108, 135, 180, 252:
  (45)      (4*27)        (135)       (4*45)        (4*63)
  (5*9)     (2*6*9)       (3*45)      (12*15)       (12*21)
  (3*15)    (3*4*9)       (5*27)      (4*5*9)       (4*7*9)
  (3*3*5)   (2*2*27)      (9*15)      (2*2*45)      (6*6*7)
            (2*2*3*9)     (3*5*9)     (2*6*15)      (2*2*63)
            (2*2*3*3*3)   (3*3*15)    (3*4*15)      (2*6*21)
                          (3*3*3*5)   (2*2*5*9)     (3*4*21)
                                      (3*3*4*5)     (2*2*7*9)
                                      (2*2*3*15)    (2*3*6*7)
                                      (2*2*3*3*5)   (3*3*4*7)
                                                    (2*2*3*21)
                                                    (2*2*3*3*7)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's are A000079.
The version for partitions is A027193.
The version for prime indices is A244991.
The version looking at length instead of greatest factor is A339890.
The version that also has odd length is A340607.
The version looking at least factor is A340832.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A058695 counts partitions of odd numbers.
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A050320, A061395, A339846, A340385, A340596, A340599, A340654, A340655, A340785, A340854, A340855.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ@*Max]],{n,100}]

A340831(n, m=n, fc=1) = if(1==n, !fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(!fc||(d%2)), s += A340831(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

A264037 Stirling transform of A077957 (aerated powers of 2) with 0 prepended [0, 1, 0, 2, 0, 4, 0, 8, ...].

Original entry on oeis.org

0, 1, 1, 3, 13, 55, 241, 1171, 6357, 37567, 236521, 1574331, 11068333, 82110535, 640794337, 5239439011, 44723250501, 397481121295, 3671081354137, 35176098791115, 349120380267421, 3583273413146647, 37975511840454673, 415004245048757299, 4670891190907818165

Offset: 0

Vladimir Reshetnikov, Nov 01 2015

nonn

a(n) without the leading zero [1, 1, 3, 13, 55, ...] is the binomial transform of A264036.

			G.f. = x + x^2 + 3*x^3 + 13*x^4 + 55*x^5 + 241*x^7 + 1171*x^8 + 6357*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..567
Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A077957, A264036.

Cf. A024429, A357572, A357598.

Table[(BellB[n, Sqrt[2]] - BellB[n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 24}]

vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k+1, 2))) \\ Altug Alkan, Nov 01 2015

a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k+1).
a(n) = (Bell_n(sqrt(2)) - Bell_n(-sqrt(2)))/(2*sqrt(2)), where Bell_n(x) is n-th Bell polynomial.
Bell_n(sqrt(2)) = A264036(n) + a(n)*sqrt(2).
E.g.f.: sinh(sqrt(2)*(exp(x) - 1))/sqrt(2).
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A264036(k). - Seiichi Manyama, Oct 12 2022

A357598 Expansion of e.g.f. sinh(2 * (exp(x)-1)) / 2.

Original entry on oeis.org

0, 1, 1, 5, 25, 117, 601, 3509, 22457, 153141, 1105561, 8453557, 68339833, 581495605, 5184047961, 48259748533, 468040609593, 4719817792565, 49396003390489, 535526127566773, 6004124908829177, 69509047405180213, 829801009239621849, 10202835010223731893

Offset: 0

Seiichi Manyama, Oct 05 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..558
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A024429, A264037, A357572.

Cf. A065143, A078944, A357599.

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(2*(exp(x)-1))/2)))

a(n) = sum(k=0, (n-1)\2, 4^k*stirling(n, 2*k+1, 2));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 2)-Bell_poly(n, -2)))/4;

a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(2) - Bell_n(-2) )/4, where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A065143(k).

cp's OEIS Frontend

A121867 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)*A(k); entry gives A sequence (cf. A121868).

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A121868 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)*A(k); entry gives B sequence (cf. A121867).

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A143817 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)*C(k). This entry gives the sequence C(n).

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A143816 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)*C(k). This entry gives the sequence B(n).

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A143818 Let R(n) = sum {k = 0..inf} (3k)^n/(3k)! for n = 0,1,2,... . Then the real number R(n) is an integral linear combination of R(0), R(1) and R(2). This sequence gives the coefficients of R(1).

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A358824 Number of twice-partitions of n of odd length.

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A121867 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)A(k); entry gives A sequence (cf. A121868).

A121868 Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)B(k), B(n+1) = Sum_{k=0..n} -binomial(n,k)A(k); entry gives B sequence (cf. A121867).

A143817 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)C(k). This entry gives the sequence C(n).

A143816 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)C(k). This entry gives the sequence B(n).