A143628 -id:A143628 - cp's OEIS frontend

A143631 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let A(n+1) = - Sum_{k = 0..n} binomial(n,k)C(k), B(n+1) = Sum_{k = 0..n} binomial(n,k)A(k) and C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k). This entry gives the sequence B(n).

0, 1, 1, 1, 0, -9, -64, -348, -1672, -7307, -28225, -81817, 14191, 3143571, 38184875, 353727284, 2916494333, 22260343389, 157677357255, 1007259846130, 5241783274713, 12146415146776, -210638381350012, -4813155361775252

Offset: 0

Peter Bala, Sep 05 2008

The other sequences are A(n) = A143628(n) and C(n) = A143630(n). Compare with A121867 and A121868. See also A143816.

Seiichi Manyama, Table of n, a(n) for n = 0..578
Eric Weisstein's World of Mathematics, Bell Polynomial.

A121867, A121868, A143628, A143629, A143630, A143816.

# Compare with A143816
#
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)*c[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)*b[k], k=0..n-1);
end do:
A143631:=[seq(b[n], n=0..M)];

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}]]; A143631 = 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 15 2022

a(n) = A143629(n) + A143630(n).
From Seiichi Manyama, Oct 15 2022: (Start)
a(n) = Sum_{k = 0..floor((n-1)/3)} (-1)^k * Stirling2(n,3*k+1).
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). (End)

A143625 Decimal expansion of the constant E_3(0) := Sum_{n >= 0} (-1)^floor(n/3)/n! = 1 + 1/1! + 1/2! - 1/3! - 1/4! - 1/5! + + + - - - ... .

Original entry on oeis.org

2, 2, 8, 4, 9, 4, 2, 3, 8, 2, 4, 0, 9, 6, 3, 5, 2, 0, 8, 9, 9, 9, 0, 5, 0, 0, 1, 9, 2, 6, 3, 0, 8, 2, 7, 0, 2, 1, 6, 1, 5, 1, 3, 2, 6, 2, 9, 9, 4, 9, 5, 8, 9, 7, 8, 5, 9, 8, 2, 8, 8, 9, 8, 0, 0, 3, 7, 3, 7, 1, 0, 1, 5, 7, 5, 1, 9, 7, 3, 4, 5, 9, 4, 0, 3, 7, 4, 4, 9, 5, 1, 2, 5, 2, 4, 6, 3, 4, 4, 8, 8

Offset: 1

Peter Bala, Aug 30 2008

Define E_3(n) = Sum_{k >= 0} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E_3(n+3) = 3*E_3(n+2) - 2*E_3(n+1) - Sum_{i = 0..n} 3^i*binomial(n,i) * E_3(n-i) for n >= 0. Thus E_3(n) is an integral linear combination of E_3(0), E_3(1) and E_3(2). See the examples below.
The decimal expansions of E_3(1) and E_3(2) are given in A143626 and A143627. Compare with A143623 and A143624.
E_3(n) as linear combination of E_3(i), i = 0..2.
=======================================
..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)
=======================================
..E_3(3)..|.....-1.......-2........3...
..E_3(4)..|.....-6.......-7........7...
..E_3(5)..|....-25......-23.......14...
..E_3(6)..|....-89......-80.......16...
..E_3(7)..|...-280.....-271......-77...
..E_3(8)..|...-700.....-750.....-922...
..E_3(9)..|...-380.....-647....-6660...
..E_3(10).|..13452....13039...-41264...
...
The columns are A143628, A143629 and A143630.

			2.284942382409635208999050...

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.

Cf. A143623, A143624, A143626, A143627, A143628, A143629, A143630.

RealDigits[ N[ (2*E^(3/2)*(Cos[Sqrt[3]/2] + Sqrt[3]*Sin[Sqrt[3]/2]) + 1)/(3*E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Offset corrected by R. J. Mathar, Feb 05 2009

A143626 Decimal expansion of the constant E_3(1) := Sum_{k >= 0} (-1)^floor(k/3)*k/k! = 1/1! + 2/2! - 3/3! - 4/4! - 5/5! + + + - - - ... .

Original entry on oeis.org

1, 3, 0, 1, 5, 5, 9, 4, 9, 5, 9, 8, 2, 9, 7, 9, 6, 0, 2, 8, 4, 3, 0, 4, 2, 7, 0, 8, 2, 5, 5, 1, 9, 9, 2, 7, 4, 2, 3, 4, 9, 4, 6, 9, 7, 2, 9, 6, 4, 7, 7, 1, 7, 0, 0, 7, 4, 7, 5, 5, 3, 4, 1, 4, 2, 0, 7, 7, 2, 4, 0, 7, 2, 9, 9, 2, 5, 4, 4, 6, 4, 4, 4, 3, 7, 4, 5, 3, 0, 1, 0, 3, 2, 0, 4, 9, 5, 8, 3, 2, 7

Offset: 1

Peter Bala, Aug 30 2008

Define E_3(n) = Sum_{k >= 0} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E_3(n+3) = 3*E_3(n+2) - 2*E_3(n+1) - Sum_{i = 0..n} 3^i*binomial(n,i) * E_3(n-i) for n >= 0. Thus E_3(n) is an integral linear combination of E_3(0), E_3(1) and E_3(2). See the examples below.
The decimal expansions of E_3(0) and E_3(2) are given in A143635 and A143627. Compare with A143623 and A143624.
E_3(n) as linear combination of E_3(i), i = 0..2.
=======================================
..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)
=======================================
..E_3(3)..|.....-1.......-2........3...
..E_3(4)..|.....-6.......-7........7...
..E_3(5)..|....-25......-23.......14...
..E_3(6)..|....-89......-80.......16...
..E_3(7)..|...-280.....-271......-77...
..E_3(8)..|...-700.....-750.....-922...
..E_3(9)..|...-380.....-647....-6660...
..E_3(10).|..13452....13039...-41264...
...
The columns are A143628, A143629 and A143630.

			1.3015594959829796028430427

Cf. A143623, A143624, A143625, A143627, A143628, A143629, A143630.

RealDigits[ N[ (4*E^(3/2)*Cos[Sqrt[3]/2] - 1)/(3*E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Offset corrected by R. J. Mathar, Feb 05 2009

A143627 Decimal expansion of the constant E_3(2) := sum {k = 0.. inf} (-1)^floor(k/3)*k^2/k! = 1/1! + 2^2/2! - 3^2/3! - 4^2/4! - 5^2/5! + + + - - - ... = 0.68605 60507 ... .

Original entry on oeis.org

6, 8, 6, 0, 5, 6, 0, 5, 0, 7, 2, 7, 7, 6, 6, 3, 1, 8, 2, 8, 2, 5, 5, 9, 1, 6, 7, 4, 0, 8, 7, 7, 6, 7, 1, 3, 7, 5, 4, 1, 9, 1, 8, 1, 3, 9, 6, 6, 3, 5, 2, 2, 5, 7, 4, 0, 4, 6, 5, 4, 6, 5, 0, 0, 7, 5, 5, 3, 8, 6, 2, 5, 9, 7, 8, 0, 1, 5, 6, 3, 2, 8, 2, 8, 3, 0, 8, 3, 4, 3, 7, 3, 4, 4, 4, 7, 8, 6, 0, 3

Offset: 0

Peter Bala, Aug 30 2008

Define E_3(n) = sum {k = 0..inf} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E_3(n+3) = 3*E_3(n+2) - 2*E_3(n+1) - sum {i = 0..n} 3^i*binomial(n,i) * E_3(n-i) for n >= 0. Thus E_3(n) is an integral linear combination of E_3(0), E_3(1) and E_3(2). See the examples below. The decimal expansions of E_3(0) and E_3(1) are given in A143625 and A143626. Compare with A143623 and A143624.

			E_3(n) as linear combination of E_3(i),
i = 0..2.
=======================================
..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)
=======================================
..E_3(3)..|.....-1.......-2........3...
..E_3(4)..|.....-6.......-7........7...
..E_3(5)..|....-25......-23.......14...
..E_3(6)..|....-89......-80.......16...
..E_3(7)..|...-280.....-271......-77...
..E_3(8)..|...-700.....-750.....-922...
..E_3(9)..|...-380.....-647....-6660...
..E_3(10).|..13452....13039...-41264...
...
The columns are A143628, A143629 and A143630.

A143623, A143624, A143625, A143626, A143628, A143629, A143630.

RealDigits[N[(8/3)*Sqrt[E]*Cos[Sqrt[3]/2] + (1/40)*(HypergeometricPFQ[{}, {7/3, 8/3}, -(1/27)] - 5*HypergeometricPFQ[{}, {5/3, 7/3}, -(1/27)]) - 2*Sqrt[E/3]*Sin[Sqrt[3]/2] - 5/(3*E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Offset corrected by R. J. Mathar, Feb 05 2009

cp's OEIS Frontend

A143631 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let A(n+1) = - Sum_{k = 0..n} binomial(n,k)*C(k), B(n+1) = Sum_{k = 0..n} binomial(n,k)*A(k) and C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k). This entry gives the sequence B(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A143625 Decimal expansion of the constant E_3(0) := Sum_{n >= 0} (-1)^floor(n/3)/n! = 1 + 1/1! + 1/2! - 1/3! - 1/4! - 1/5! + + + - - - ... .

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Extensions

A143626 Decimal expansion of the constant E_3(1) := Sum_{k >= 0} (-1)^floor(k/3)*k/k! = 1/1! + 2/2! - 3/3! - 4/4! - 5/5! + + + - - - ... .

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A143627 Decimal expansion of the constant E_3(2) := sum {k = 0.. inf} (-1)^floor(k/3)*k^2/k! = 1/1! + 2^2/2! - 3^2/3! - 4^2/4! - 5^2/5! + + + - - - ... = 0.68605 60507 ... .

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A143631 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let A(n+1) = - Sum_{k = 0..n} binomial(n,k)C(k), B(n+1) = Sum_{k = 0..n} binomial(n,k)A(k) and C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k). This entry gives the sequence B(n).