A166895 -id:A166895 - cp's OEIS frontend

A166894 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^n * x^k] * x^n/n ), an integer series in x.

1, 1, 2, 4, 14, 89, 1050, 28983, 2066217, 272159513, 56735786726, 23441305184736, 26635730598676118, 64099902414443754551, 241666593661232949435382, 1531373212165249576810266758, 24642808245610936988728333582900

Offset: 0

Paul D. Hanna, Nov 23 2009

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 89*x^5 + 1050*x^6 +...
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 366*x^5/5 + 5697*x^6/6 +...+ A166895(n)*x^n/n +...

Cf. A166895.

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^m*x^k)*x^m/m)+x*O(x^n)), n)}

{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m\2, binomial(m-k, k)^(m-k)*m/(m-k))*x^m/m)+x*O(x^n)), n)}

G.f.: exp( Sum_{n>=1} A166895(n)*x^n/n ) where A166895(n) = Sum_{k=0..[n/2]} C(n-k,k)^(n-k)*n/(n-k).

A166897 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^3*n/(n-k), n>=1.

Original entry on oeis.org

1, 3, 13, 39, 126, 477, 1765, 6495, 24709, 95128, 367368, 1431453, 5620343, 22170543, 87858813, 349708431, 1397003136, 5598513261, 22502171771, 90681323364, 366299212873, 1482827487650, 6014529069540, 24439715146941

Offset: 1

Paul D. Hanna, Nov 23 2009

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 39*x^4/4 + 126*x^5/5 + 477*x^6/6 +...
exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...+ A166896(n)*x^n/n +...

Cf. A166897, variants: A167539, A166895, A166899.

Table[Sum[Binomial[n-k,k]^3 n/(n-k),{k,0,Floor[n/2]}],{n,30}] (* Harvey P. Dale, Mar 05 2013 *)

a(n)=sum(k=0,n\2,binomial(n-k,k)^3*n/(n-k))

Logarithmic derivative of A166896.

a(n) ~ sqrt(15) * phi^(3*n + 2) / (6*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 27 2017

A181081 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n-2k+1) n/(n-k).

Original entry on oeis.org

1, 3, 7, 39, 336, 4077, 68461, 1955295, 129385141, 17371664728, 3501431925168, 947675920795833, 355261887514210899, 236156938257380344851, 390707976511340699319417, 1324768245535417597286345871

Offset: 1

Paul D. Hanna, Oct 02 2010

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 336*x^5/5 + ...
which equals the series:
log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^3*x + 3^2*x^2 + x^3)*x^3/3
+ (1+ 4^4*x + 6^3*x^2 + 4^2*x^3 + x^4)*x^4/4
+ (1+ 5^5*x + 10^4*x^2 + 10^3*x^3 + 5^2*x^4 + x^5)*x^5/5
+ (1+ 6^6*x + 15^5*x^2 + 20^4*x^3 + 15^3*x^4 + 6^2*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181080:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 + 10641*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 1..90

Cf. A181080 (exp), variants: A181071, A166895.

[(&+[Binomial(n-j,j)^(n-2*j+1)*(n/(n-j)): j in [0..Floor(n/2)]]): n in [1..20]]; // G. C. Greubel, Apr 04 2021

Table[Sum[Binomial[n-k, k]^(n-2*k+1)*(n/(n-k)), {k, 0, Floor[n/2]}], {n, 20}] (* G. C. Greubel, Apr 04 2021 *)

a(n)=sum(k=0, n\2, binomial(n-k, k)^(n-2*k+1)*n/(n-k))

{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m-k+1)*x^k)*x^m/m)+x*O(x^n), n)}

[sum( binomial(n-k, k)^(n-2*k+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # G. C. Greubel, Apr 04 2021

L.g.f.: L(x) = Sum_{n>=1} ( Sum_{k=0..n} binomial(n,k)^(n-k+1)*x^k ) * x^n/n.

Logarithmic derivative of A181080.

A323777 Decimal expansion of the root of the equation (1-2r)^(3-4r) = (1-r)^(2-2r) r^(1-2*r).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 27 2019

			0.2206760413237406963128222699980501671876810310275740395417335127215630565...

Cf. A166895, A209331, A209428.

Cf. A220359, A237421, A245259, A323768, A323773, A323778.

RealDigits[r/.FindRoot[(1-2*r)^(3-4*r) == (1-r)^(2-2*r) * r^(1-2*r), {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A166899 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^4*n/(n-k), n>=1.

Original entry on oeis.org

1, 3, 25, 111, 456, 2697, 15961, 86247, 495781, 3003738, 17946798, 107667969, 660458787, 4081397547, 25274724105, 157744019799, 991384251102, 6254115981009, 39613066988527, 252017709962526, 1608980424431755

Offset: 1

Paul D. Hanna, Nov 23 2009

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 111*x^4/4 + 456*x^5/5 + 2697*x^6/6 +...
exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 38*x^4 + 137*x^5 + 646*x^6 + 3241*x^7 +...+ A166898(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 1..501 [Offset shifted by _Georg Fischer_, Nov 20 2024]

Cf. A166898, variants: A167539, A166895, A166897.

Table[Sum[Binomial[n - k, k]^4 *n/(n - k), {k, 0, Floor[n/2]}], {n, 1, 50}] (* G. C. Greubel, May 27 2016 *)

a(n)=sum(k=0,n\2,binomial(n-k,k)^4*n/(n-k))

Logarithmic derivative of A166898.

a(n) ~ 5^(3/4) * phi^(4*n+3) / (2^(5/2) * Pi^(3/2) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 27 2017

Offset changed to 1 by Georg Fischer, Nov 20 2024

A167539 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^2 * n/(n-k), n>=1.

Original entry on oeis.org

1, 3, 7, 15, 36, 87, 211, 519, 1285, 3198, 7998, 20079, 50571, 127725, 323367, 820407, 2085306, 5309169, 13537045, 34561890, 88347091, 226079208, 579110262, 1484766015, 3809948461, 9783998877, 25143452881, 64658016249, 166375274790

Offset: 1

Paul D. Hanna, Nov 23 2009

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 36*x^5/5 + 87*x^6/6 +...
exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...+ A004148(n+1)*x^n/n +...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A004148, variants: A166895, A166897, A166899.

Table[Sum[(Binomial[n - k, k]^2)*(n/(n - k)), {k, 0, n/2}], {n, 1, 100}] (* G. C. Greubel, Jun 15 2016 *)

{a(n) = sum(k=0,n\2, binomial(n-k,k)^2 * n/(n-k))}
for(n=1,30,print1(a(n),", "))

{a(n) = n * polcoeff( log( (1 - x - x^2 - sqrt((1+x+x^2)*(1-3*x+x^2) +x^6*O(x^n) )) / (2*x^3) ), n)}
for(n=1,30,print1(a(n),", "))

{a(n) = sum(k=0,n-1,sum(j=0,k, binomial(n-k+j,n-k)*n/(n-k+j) * binomial(n-k,k-j)*binomial(k-j,j)))}
for(n=1,30,print1(a(n),", "))

L.g.f.: Log((1 - x - x^2 - sqrt((1+x+x^2)*(1-3*x+x^2)))/(2*x^3)) = Sum_{n>=1} a(n)*x^n/n. - Paul D. Hanna, Jul 19 2015
L.g.f.: -Log((1 - x - x^2 + sqrt((1+x+x^2)*(1-3*x+x^2)))/2) = Sum_{n>=1} a(n)*x^n/n. (Minor simplification of the l.g.f. given above.) - Petros Hadjicostas, Oct 25 2017
a(n) = Sum_{k=0..n-1} Sum_{j=0..k} C(n-k+j,n-k)*n/(n-k+j) * C(n-k,k-j)*C(k-j,j).
a(n) ~ 5^(1/4) * phi^(2*n + 1) / (2*sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 27 2017

A181083 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n * n/(n-k).

Original entry on oeis.org

1, 3, 13, 111, 1686, 88737, 14355265, 3583775847, 1789371713317, 4311992850152298, 23667113846872049808, 185391762466214524964649, 4305238471804328835068596175, 468653724243371951619336632177235

Offset: 1

Paul D. Hanna, Oct 02 2010

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 111*x^4/4 + 1686*x^5/5 + ...
which equals the series:
  L(x) = (1 + x)*x + (1 + 2^3*x + x^2)*x^2/2
  + (1 + 3^4*x +  3^5*x^2 +      x^3)*x^3/3
  + (1 + 4^5*x +  6^6*x^2 +  4^7*x^3 +       x^4)*x^4/4
  + (1 + 5^6*x + 10^7*x^2 + 10^8*x^3 +   5^9*x^4 +      x^5)*x^5/5
  + (1 + 6^7*x + 15^8*x^2 + 20^9*x^3 + 15^10*x^4 + 6^11*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181082:
  exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 34*x^4 + 375*x^5 + 15200*x^6 + 2066401*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 1..70

Variants: A166895, A181071, A181081.

Cf. A181082 (exp).

[(&+[Binomial(n-k,k)^n*(n/(n-k)): k in [0..Floor(n/2)]]): n in [1..25]]; // G. C. Greubel, Apr 05 2021

A181083:= n-> add(binomial(n-k,k)^n*(n/(n-k)), k=0..floor(n/2)); seq(A181083(n), n=1..20); # G. C. Greubel, Apr 05 2021

Table[Sum[Binomial[n-k,k]^n n/(n-k),{k,0,Floor[n/2]}],{n,20}] (* Harvey P. Dale, Jun 24 2015 *)

a(n)=sum(k=0, n\2, binomial(n-k, k)^n*n/(n-k))

{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k)*x^k)*x^m/m)+x*O(x^n), n)}

[sum(binomial(n-k,k)^n*(n/(n-k)) for k in (0..n//2)) for n in (1..25)] # G. C. Greubel, Apr 05 2021

L.g.f.: L(x) = Sum_{n>=1} [Sum_{k=0..n} binomial(n,k)^(n+k)*x^k] * x^n/n.

Logarithmic derivative of A181082.

A181085 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n+1) * n/(n-k).

Original entry on oeis.org

1, 3, 25, 327, 6336, 513657, 142074241, 52903930911, 36806786795365, 148308705637730728, 1318954828711012426638, 15279013243159345043036553, 534104982404807772659968455891, 97749134742042348389685885848315523

Offset: 1

Paul D. Hanna, Oct 28 2010

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + ...
which equals the series:
L(x) = (1 + x)*x + (1 + 2^4*x + x^2)*x^2/2
+ (1+ 3^5*x + 3^6*x^2 + x^3)*x^3/3
+ (1+ 4^6*x + 6^7*x^2 + 4^8*x^3 + x^4)*x^4/4
+ (1+ 5^7*x + 10^8*x^2 + 10^9*x^3 + 5^10*x^4 + x^5)*x^5/5
+ (1+ 6^8*x + 15^9*x^2 + 20^10*x^3 + 15^11*x^4 + 6^12*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181084:
exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 1..60

Variants: A166895, A181071, A181081, A181083.

Cf. A181084 (exp).

[(&+[Binomial(n-j,j)^(n+1)*(n/(n-j)): j in [0..Floor(n/2)]]): j in [1..20]]; // G. C. Greubel, Apr 04 2021

Table[Sum[Binomial[n-k, k]^(n+1)*(n/(n-k)), {k, 0, Floor[n/2]}], {n, 20}] (* G. C. Greubel, Apr 04 2021 *)

a(n)=sum(k=0, n\2, binomial(n-k, k)^(n+1)*n/(n-k))

{a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k+1)*x^k)*x^m/m)+x*O(x^n), n)}

[sum( binomial(n-k, k)^(n+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # G. C. Greubel, Apr 04 2021

cp's OEIS Frontend

A166894 G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^n * x^k] * x^n/n ), an integer series in x.

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A166897 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^3*n/(n-k), n>=1.

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A181081 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n-2k*+1) * n/(n-k).

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A323777 Decimal expansion of the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r).

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A166899 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^4*n/(n-k), n>=1.

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A167539 a(n) = Sum_{k=0..[n/2]} C(n-k,k)^2 * n/(n-k), n>=1.

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A181083 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n * n/(n-k).

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A181081 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n-2k+1) n/(n-k).

A323777 Decimal expansion of the root of the equation (1-2r)^(3-4r) = (1-r)^(2-2r) r^(1-2*r).