A167539 -id:A167539 - cp's OEIS frontend

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

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

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

A123612 Antidiagonal sums of triangle A123610.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 17, 31, 68, 145, 325, 728, 1685, 3891, 9140, 21565, 51311, 122666, 295037, 712477, 1728262, 4207027, 10276693, 25178708, 61866141, 152397945, 376309596, 931239093, 2309219447, 5737078442, 14278587533, 35595622719

Offset: 0

Paul D. Hanna, Oct 03 2006

nonn

The g.f. was suggested by P. D. Hanna. It can be proved either by letting y=x in the bivariate g.f. for sequence A123610 or by using the formula of A. Howroyd (below) for this sequence and the l.g.f. for sequence A167539. The second proof proceeds as follows: Sum_{n>=1} a(n)*x^n = Sum_{n>=1} (1/n)*Sum_{d|n} phi(n/d)*g(d), where g(d) = A167539(d). Then Sum_{n>=1} a(n)*x^n = Sum_{m>=1} (phi(m)/m)*Sum_{d>=1} g(d)*(x^m)^d/d = Sum_{m>=1} (phi(m)/m)*G(x^m), where G(x) = l.g.f. of sequence g(n) = A167539(n). - Petros Hadjicostas, Oct 25 2017

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A123610 (triangle), A123611 (row sums); central terms: A123617, A123618, A167539.

Total /@ Table[Function[m, If[k == 0, 1, 1/m DivisorSum[m, If[GCD[k, #] == #, EulerPhi[#] Binomial[m/#, k/#]^2, 0] &]]][n - k + 1], {n, -1, 30}, {k, 0, Ceiling[n/2]}] (* Michael De Vlieger, Apr 03 2017, after Jean-François Alcover at A123610 *)

{a(n)=sum(k=0,n\2,if(k==0,1,(1/(n-k))*sumdiv(n-k,d,if(gcd(k,d)==d, eulerphi(d)*binomial((n-k)/d,k/d)^2,0))))}

a(n) = (1/n) * Sum_{d | n} phi(n/d) * A167539(d) for n>0. - Andrew Howroyd, Apr 02 2017

G.f.: 1-Sum_{n>=1} (phi(n)/n)*f(x^n), where f(x) = log((1-x-x^2+sqrt((1+x+x^2)*(1-3*x+x^2)))/2) = -log((1-x-x^2-sqrt((1+x+x^2)*(1-3*x+x^2)))/(2*x^3)). - Petros Hadjicostas, Oct 25 2017

cp's OEIS Frontend

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

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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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A123612 Antidiagonal sums of triangle A123610.

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