A096048 -id:A096048 - cp's OEIS frontend

A096045 a(n) = B(2n, 2)/B(2n) (see formula section).

1, 10, 46, 190, 766, 3070, 12286, 49150, 196606, 786430, 3145726, 12582910, 50331646, 201326590, 805306366, 3221225470, 12884901886, 51539607550, 206158430206, 824633720830, 3298534883326, 13194139533310, 52776558133246, 211106232532990, 844424930131966

Offset: 0

Benoit Cloitre, Jun 17 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000302, A002063, A096046, A096047, A096048.

[3*4^n-2: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

a[n_]:= Sum[2^k*Sum[Binomial[2*n, j]*BernoulliB[j], {j,0,k}], {k,0,2*n}]/BernoulliB[2*n]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jan 14 2015 *)
NestList[4#+6&,1,30] (* Harvey P. Dale, Dec 27 2016 *)

A096045(n):=3*4^n-2$ makelist(A096045(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

a(n)=sum(i=0,2*n,2^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n)

[3*4^n-2 for n in range(41)] # G. C. Greubel, Jan 22 2023

a(n) = B(2*n, 2)/B(2*n), where B(n, p) = Sum_{i=0..n} p^i * (Sum_{j=0..i} binomial(n,j)*B(j)) with B(k) = k-th Bernoulli number.
a(n) = 3*4^n - 2.
a(n) = 5*a(n-1) - 4*a(n-2).
a(n) = 4*a(n-1) + 6. First differences give A002063. - Paul Curtz, Jul 07 2008
From G. C. Greubel, Jan 22 2023: (Start)
a(n) = 3*A000302(n) - 2.
G.f.: (1+5*x)/((1-x)*(1-4*x)).
E.g.f.: 3*exp(4*x) - 2*exp(x). (End)

A096046 a(n) = B(2n,3)/B(2n) (see comment).

Original entry on oeis.org

1, 15, 141, 1275, 11481, 103335, 930021, 8370195, 75331761, 677985855, 6101872701, 54916854315, 494251688841, 4448265199575, 40034386796181, 360309481165635, 3242785330490721, 29185067974416495, 262665611769748461

Offset: 0

Benoit Cloitre, Jun 17 2004

nonn

B(n,p) = Sum_{i=0..n} p^i*Sum_{j=0..i} binomial(n,j)*B(j) where B(k) = k-th Bernoulli number.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A096045, A096047, A096048.

[(1/4)*(7*9^n-3): n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

A096046(n):=(1/4)*(7*9^n-3)$ makelist(A096046(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

a(n)=sum(i=0,2*n,3^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n)

a(n) = (1/4)*(7*9^n - 3).
a(n) = 10*a(n-1) - 9*a(n-2); a(0)=1, a(1)=15.
a(n) = 9*a(n-1) + 6. First differences = 14*A001019(n). - Paul Curtz, Jul 07 2008

A096047 a(n)=B(2n,4)/B(2n) (see comment).

Original entry on oeis.org

1, 22, 346, 5482, 87466, 1398442, 22370986, 357919402, 5726644906, 91626056362, 1466015853226, 23456249457322, 375299974539946, 6004799525530282, 96076792140049066, 1537228673167043242, 24595658766377724586

Offset: 0

Benoit Cloitre, Jun 17 2004

B(n,p)=sum(i=0,n,p^i*sum(j=0,i,binomial(n,j)*B(j))) where B(k)=k-th Bernoulli number

Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A096045, A096046, A096048.

LinearRecurrence[{21,-84,64},{1,22,346},20] (* Harvey P. Dale, Oct 13 2016 *)

a[0]:1$ a[1]:22$ a[2]:346$ a[n]:=(1/3)*(4*16^n+4^n-2)$ A096047(n):=a[n]$ makelist(A096047(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

a(n)=sum(i=0,2*n,4^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n)

a(n)=(1/3)*(4*16^n+4^n-2); a(0)=1, a(1)=22, a(2)=346 and a(n)=21*a(n-1)-84*a(n-2)+64*a(n-3)

A096049 a(n) = [B(2n,5)/B(2n)] ( [x] = floor(x), see comment for B(n,k) definition ).

Original entry on oeis.org

1, 31, 745, 18397, 458545, 11455304, 286331664, 7157976493, 178947452208, 4473674081283, 111841775707840, 2796043915880138, 69901094917491465, 1747527354316971026, 43688183741551848165, 1092204592811481165247, 27305114815741345242261, 682627870365123204281633

Offset: 0

Benoit Cloitre, Jun 17 2004

nonn

B(n,p) = Sum_{i=0..n} p^i*(Sum_{j=0..i} binomial(n,j)*B(j)) where B(k)=k-th Bernoulli number. B(2n,p)/B(2n) take integer values for all n if p=1,2,3,4,6. p=5 is the smallest integer for which B(2n,5)/B(2n) is not always integer valued.

Cf. A096045, A096046, A096047, A096048.

a(n)=floor(sum(i=0,2*n,5^i*sum(j=0,i,binomial(2*n,j)*bernfrac(j)))/bernfrac(2*n))

a(n) = floor((1/16)*(21-sqrt(5))*25^n + (1/8)*sqrt(5)*((25/4)^n+(25/9)^n-(25/16)^n) - (1/16)*(5-sqrt(5)) + (1/4)*sqrt(5)*(25/36)^n).

A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.

Original entry on oeis.org

1, 91, 3751, 138811, 5028751, 181308931, 6529545751, 235085301451, 8463265086751, 304679288612371, 10968470088963751, 394865064451017691, 14215143591303768751, 511745180725868773411, 18422826609078989373751, 663221758853362301815531, 23875983327059668074930751

Offset: 1

Benoit Cloitre, Jun 18 2004

Colin Barker, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (50,-553,1800,-1296).

Cf. A083420, A096045, A096046, A096047, A096048, A096053.

a[n_] := 6^(2*n-1) * BernoulliB[2*n, 1/6] / BernoulliB[2*n]; Array[a, 15] (* Amiram Eldar, May 07 2025 *)

a(n)=(1/12)*36^n-(1/6)*9^n-(1/4)*4^n+1/2;

a(n) = (1/12)*(36^n - 2*9^n - 3*4^n+6).
From Colin Barker, May 30 2020: (Start)
G.f.: x*(1 - 6*x)*(1 + 47*x + 36*x^2) / ((1 - x)*(1 - 4*x)*(1 - 9*x)*(1 - 36*x)).
a(n) = 50*a(n-1) - 553*a(n-2) + 1800*a(n-3) - 1296*a(n-4) for n>4. (End)

A096050 Decimal expansion of lim_{n->oo} B(2n,7)/(B(2n)*49^n) (see comment for B(n,k) definition).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jun 17 2004

B(n,p) = Sum_{i=0..n} p^i * Sum_{j=0..i} binomial(n,j)*B(j) where B(k) = k-th Bernoulli number. B(2n,p)/B(2n) takes integer values for all n if p=1,2,3,4,6. p=5 is the smallest integer for which B(2n,5)/B(2n) is not always integer-valued. And lim_{n->oo} B(2n,5)/(B(2n)*25^n) = (21-sqrt(5))/16.

Cf. A096045, A096046, A096047, A096048, A096049.

RealDigits[x/.FindRoot[ 1728x^3-6192x^2+7368x-2911==0,{x,1}, WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Feb 19 2012 *)

solve(q=1,1.1,1728*q^3-6192*q^2+7368*q-2911)

Limit_{n->oo} B(2n, 7)/(B(2n)*49^n) = 1.0627516996902110782... is the smallest root of 1728*X^3 - 6192*X^2 + 7368*X - 2911 = 0.

A096051 Decimal expansion of lim_{n->oo} B(2n,8)/(B(2n)*64^n) (see comment for B(n,k) definition).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jun 17 2004

B(n,p) = Sum_{i=0..n} (p^i * Sum_{j=0..i} binomial(n,j)*B(j)) where B(k) is the k-th Bernoulli number.

			1.04184188840192178222845080541359299438788058033021...

Cf. A027641, A027642, A096045, A096046, A096047, A096048, A096049, A096050.

RealDigits[(16 - Sqrt[2])/14, 10, 100][[1]] (* Amiram Eldar, May 08 2022 *)

PARI
```
(16-sqrt(2))/14
```

Equals (16-sqrt(2))/14.

A096052 Decimal expansion of lim_{n->oo} B(2n,5)/(B(2n)*25^n) (see comment for B(n,k) definition).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jun 17 2004

B(n,p) = Sum_{i=0..n} (p^i * Sum_{j=0..i} binomial(n,j)*B(j)) where B(k) is the k-th Bernoulli number.

			1.17274575140626314397442664570429523528496135252427...

Cf. A027641, A027642, A096045, A096046, A096047, A096048, A096049, A096050, A096051.

RealDigits[(21 - Sqrt[5])/16, 10, 100][[1]] (* Amiram Eldar, May 08 2022 *)

PARI
```
(21-sqrt(5))/16
```

Equals (21-sqrt(5))/16.

cp's OEIS Frontend

A096045 a(n) = B(2*n, 2)/B(2*n) (see formula section).

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A096046 a(n) = B(2n,3)/B(2n) (see comment).

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A096047 a(n)=B(2n,4)/B(2n) (see comment).

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A096049 a(n) = [B(2n,5)/B(2n)] ( [x] = floor(x), see comment for B(n,k) definition ).

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A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.

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A096050 Decimal expansion of lim_{n->oo} B(2n,7)/(B(2n)*49^n) (see comment for B(n,k) definition).

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A096051 Decimal expansion of lim_{n->oo} B(2n,8)/(B(2n)*64^n) (see comment for B(n,k) definition).

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A096052 Decimal expansion of lim_{n->oo} B(2n,5)/(B(2n)*25^n) (see comment for B(n,k) definition).

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A096045 a(n) = B(2n, 2)/B(2n) (see formula section).