A204062 -id:A204062 - cp's OEIS frontend

A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.

1, 1, 5, 21, 101, 501, 2561, 13345, 70561, 377281, 2035285, 11059205, 60454005, 332138405, 1832677185, 10150115201, 56398558081, 314273655745, 1755700634981, 9830544087221, 55155558312901, 310027473436821, 1745567243959105, 9843160519978401, 55582528404717601

Offset: 0

Paul D. Hanna, Jan 10 2012

nonn

a(n) == 1 (mod 5) iff n has no 2's in its base 5 expansion (A023729), otherwise a(n) == 0 (mod 5); this is a conjecture needing proof.

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 101*x^4 + 501*x^5 + 2561*x^6 +...
where log(A(x)) = x + 3^2*x^2/2 + 7^2*x^3/3 + 17^2*x^4/4 + 41^2*x^5/5 + 99^2*x^6/6 + 239^2*x^7/7 +...+ A001333(n)^2*x^n/n +...
The last digit of the terms in this sequence seems to be either a '1' or a '5':
by conjecture, a(n) == 0 (mod 5) whenever n has a 2 in its base 5 expansion;
if true, terms a(2*5^k) through a(3*5^k - 1) all end with digit '5' for k>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..625

Cf. A204062, A026933, A001333, A023729.

{A001333(n)=polcoeff((1-x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A001333(k)^2*x^k/k)+x*O(x^n)), n)}

{a(n)=polcoeff(1/(sqrt(1+x+x*O(x^n))*(1-6*x+x^2+x*O(x^n))^(1/4)),n)}

G.f.: 1 / ( sqrt(1+x) * (1-6*x+x^2)^(1/4) ).
Self-convolution yields A026933: Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers.
a(n) ~ 2^(1/8) * GAMMA(3/4) * (3+2*sqrt(2))^(n+1/2) / (4 * Pi * n^(3/4)). - Vaclav Kotesovec, Oct 30 2014

A026933 Self-convolution of array T given by A008288.

Original entry on oeis.org

1, 2, 11, 52, 269, 1414, 7575, 41064, 224665, 1237898, 6859555, 38187164, 213408805, 1196524814, 6727323439, 37915058384, 214140178225, 1211694546194, 6867622511675, 38981807403268, 221562006394173, 1260814207833750, 7182599953332423, 40958645048598840, 233779564099963081

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 15.

Cf. A002203, A008288, A204061, A204062.

Table[SeriesCoefficient[1/(1+x)/Sqrt[1-6*x+x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[ n_]:= Sum[ SeriesCoefficient[ SeriesCoefficient[1/(1-x-y-x*y) , {x,0,n-k}] , {y, 0, k}]^2, {k, 0, n}]; (* Michael Somos, Jun 27 2017 *)
A026933[n_]:= Sum[(Binomial[n, k]*Hypergeometric2F1[-k,k-n,-n,-1])^2, {k,0,n}];
Table[A026933[n], {n, 0, 40}] (* G. C. Greubel, May 25 2021 *)

/* Sum of squares of Delannoy numbers: */
{a(n)=sum(k=0,n,polcoeff(polcoeff(1/(1-x-y-x*y +x*O(x^n)+y*O(y^k)),n-k,x),k,y)^2)} \\ Paul D. Hanna, Jan 10 2012

/* Involving squares of companion Pell numbers: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2/2*x^k/k)+x*O(x^n)), n)}
\\ Paul D. Hanna, Jan 10 2012

my(x='x+O('x^66)); Vec( 1/(1+x)/sqrt(1-6*x+x^2) ) \\ Joerg Arndt, May 04 2013

def A026933_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*sqrt(1-6*x+x^2)) ).list()
A026933_list(40) # G. C. Greubel, May 25 2021

a(n) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers. - Paul D. Hanna, Jan 10 2012
G.f.: 1/((1+x)*sqrt(1-6*x+x^2)). - Vladeta Jovovic, May 13 2003
a(n) = (-1)^n*Sum_{k=0...n} (-1)^k*A001850(k). - Benoit Cloitre, Sep 28 2005
G.f.: exp( Sum_{n>=1} A002203(n)^2/2 * x^n/n ), where A002203 are the companion Pell numbers. - Paul D. Hanna, Jan 10 2012
Self-convolution yields A204062; self-convolution of A204061. - Paul D. Hanna, Jan 10 2012
From Vaclav Kotesovec, Oct 08 2012: (Start)
Recurrence: n*a(n) = (5*n-3)*a(n-1) + (5*n-2)*a(n-2) - (n-1)*a(n-3).
a(n) ~ sqrt(24+17*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). (End)
0 = +a(n)*(+a(n+1) -8*a(n+2) -7*a(n+3) +2*a(n+4)) +a(n+1)*(-2*a(n+1) +22*a(n+2) +20*a(n+3) -7*a(n+4)) +a(n+2)*(+30*a(n+2) +22*a(n+3) -8*a(n+4)) +a(n+3)*(-2*a(n+3) +a(n+4)) for all n in Z. - Michael Somos, Jun 27 2017

More terms from Vladeta Jovovic, May 13 2003

A212442 G.f.: exp( Sum_{n>=1} A002203(n)^3 * x^n/n ), where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 8, 140, 1864, 26602, 373080, 5253564, 73911192, 1040045475, 14634444720, 205922568360, 2897549559600, 40771618763540, 573700205699920, 8072574516567400, 113589743388536528, 1598328982089075749, 22490195492277648120, 316461065874934143252

Offset: 0

Paul D. Hanna, May 17 2012

nonn

More generally, exp(Sum_{k>=1} A002203(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A002203(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

Compare to g.f. exp(Sum_{k>=1} A002203(k) * x^k/k) = 1/(1-2*x-x^2).

			G.f.: A(x) = 1 + 8*x + 140*x^2 + 1864*x^3 + 26602*x^4 + 373080*x^5 + ...
where
log(A(x)) = 2^3*x + 6^3*x^2/2 + 14^3*x^3/3 + 34^3*x^4/4 + 82^3*x^5/5 + 198^3*x^6/6 + 478^3*x^7/7 + 1154^3*x^8/8 + ... + A002203(n)^3*x^n/n + ...
Also, the g.f. equals the infinite product:
A(x) = 1/( (1-2*x-x^2)^4 * (1-6*x^2+x^4)^16 * (1-14*x^3-x^6)^64 * (1-34*x^4+x^8)^280 * (1-82*x^5-x^10)^1344 * (1-198*x^6+x^12)^6496 * ... * (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) * ...).
The exponents in these products begin:
A212443 = [4, 16, 64, 280, 1344, 6496, 32640, 166320, 862400, ...].
The companion Pell numbers begin (at offset 1):
A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].

G. C. Greubel, Table of n, a(n) for n = 0..865
Index entries for linear recurrences with constant coefficients, signature (8,76,136,-38,-136,76,-8,-1).

Cf. A212443, A203803, A002203, A204062.

CoefficientList[Series[1/((1+2x-x^2)^3(1-14x-x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{8,76,136,-38,-136,76,-8,-1},{1,8,140,1864,26602,373080,5253564,73911192},30] (* Harvey P. Dale, Feb 15 2015 *)

/* Subroutine for the PARI programs that follow: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}

/* G.F. by Definition: */
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^3*x^k/k)+x*O(x^n)), n)}

/* G.F. as a Finite Product: */
{a(n, m=1)=polcoeff(prod(k=0, m, 1/(1 - (-1)^(m-k)*A002203(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1, m-k)), n)}

/* G.F. as an Infinite Product: */
{A212443(n)=(1/n)*sumdiv(n,d, moebius(n/d)*A002203(d)^2)}
{a(n)=polcoeff(1/prod(m=1,n, (1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))^A212443(m)),n)}
for(n=0,30,print1(a(n),", "))

G.f.: 1 / ( (1+2*x-x^2)^3 * (1-14*x-x^2) ).
G.f.: 1 / Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) where A212443(n) = (1/n)*Sum_{d|n} moebius(n/d)*A002203(d)^2.
a(0)=1, a(1)=8, a(2)=140, a(3)=1864, a(4)=26602, a(5)=373080, a(6)=5253564, a(7)=73911192, a(n) = 8*a(n-1) + 76*a(n-2) + 136*a(n-3) - 38*a(n-4) - 136*a(n-5) + 76*a(n-6) - 8*a(n-7) - a(n-8). - Harvey P. Dale, Feb 15 2015

A208034 G.f.: exp( Sum_{n>=1} 2Pell(n)^2 x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 6, 26, 122, 602, 3062, 15906, 83906, 447842, 2412566, 13094490, 71513210, 392592410, 2164815590, 11982792386, 66548673282, 370672213826, 2069974290726, 11586244722202, 64986102400122, 365183031749722, 2055594717395926, 11588727763937506, 65425688924696002

Offset: 0

Paul D. Hanna, Feb 22 2012

nonn

Conjectures: For all positive integers k,
(1) exp( Sum_{n>=1} 2*Pell(n)^(2*k) * x^n/n ) is an integer series;
(2) exp( Sum_{n>=1} 2*Pell(n)^(2*k-1) * x^n/n ) is NOT an integer series;
(3) exp( Sum_{n>=1} Pell(n)^(2*k) * x^n/n ) is NOT an integer series.

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 26*x^3 + 122*x^4 + 602*x^5 + 3062*x^6 + ...
such that, by definition,
log(A(x))/2 = x + 2^2*x^2/2 + 5^2*x^3/3 + 12^2*x^4/4 + 29^2*x^5/5 + 70^2*x^6/6 + 169^2*x^7/7 + 408^2*x^8/8 + ... + Pell(n)^2*x^n/n + ...

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

Cf. A000129, A208055, A208056, A204061, A204062.

CoefficientList[Series[Sqrt[1 + x] / (1 - 6 x + x^2)^(1/4), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 26 2018 *)

{Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^2*x^m/m) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

seq(n)={Vec(sqrt(1 + x + O(x^n)) / sqrt(sqrt(1 - 6*x + x^2 + O(x^n))))} \\ Andrew Howroyd, Feb 25 2018

G.f.: sqrt(1 + x) / (1 - 6*x + x^2)^(1/4).

a(n) ~ 2^(1/8) * (1 + sqrt(2))^(2*n) / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 31 2024

A208056 G.f.: exp( Sum_{n>=1} 2Pell(n)^(2n) * x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 18, 10450, 215011842, 168283323489554, 4613762736903044410402, 4429409381416783893511092430530, 147401742703370819998531165821635082467298, 169293247178836261713452084817353169649400098579929282

Offset: 0

Paul D. Hanna, Feb 22 2012

nonn

Given g.f. A(x), note that A(x)^(1/2) does not yield an integer series.

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 10450*x^3 + 215011842*x^4 +...
such that, by definition,
log(A(x))/2 = x + 2^4*x^2/2 + 5^6*x^3/3 + 12^8*x^4/4 + 29^10*x^5/5 + 70^12*x^6/6 + 169^14*x^7/7 +...+ Pell(n)^(2*n)*x^n/n +...

Cf. A000129, A211891, A208034, A208055, A204061, A204062.

{Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^(2*m)*x^m/m) +x*O(x^n)),n)}
for(n=0,15,print1(a(n),", "))

A208055 G.f.: exp( Sum_{n>=1} 2Pell(n)^4 x^n/n ), where Pell(n) = A000129(n).

Original entry on oeis.org

1, 2, 18, 450, 11362, 311426, 8857426, 259072706, 7730804098, 234255654466, 7184570715602, 222512186923010, 6947171244623714, 218374183252085826, 6903938704875627410, 219355658720815861378, 6999679608428089841154, 224210965624588803552642

Offset: 0

Paul D. Hanna, Feb 22 2012

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 450*x^3 + 11362*x^4 + 311426*x^5 +...
such that, by definition,
log(A(x))/2 = x + 2^4*x^2/2 + 5^4*x^3/3 + 12^4*x^4/4 + 29^4*x^5/5 + 70^4*x^6/6 + 169^4*x^7/7 + 408^4*x^8/8 +...+ Pell(n)^4*x^n/n +...

Cf. A000129, A208034, A208056, A204061, A204062, A207969.

{Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^4*x^m/m) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

The o.g.f. A(x) = 1 + 2*x + 18*x^2 + 450*x^3 + ... is an algebraic function: A(x)^32 = (1 + 6*x + x^2)^4/( (1 - 34*x + x^2)*(1 - 2*x + x^2)^3 ). Cf. A207969. - Peter Bala, Apr 03 2014
From Vaclav Kotesovec, Oct 31 2024: (Start)
G.f.: (1 + x*(6 + x))^(1/8) / ((1 - x)^(3/16)*(1 + (-17 + 12*sqrt(2))*x)^(1/32) * (1 - (17 + 12*sqrt(2))*x)^(1/32)).
a(n) ~ 5^(1/8) * (1 + sqrt(2))^(4*n) / (2^(13/64) * 3^(1/32) * Gamma(1/32) * n^(31/32)). (End)

A215171 G.f.: exp( Sum_{n>=1} A002203(n)^4 * x^n/n ), where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 16, 776, 23856, 834596, 28135056, 957599096, 32515276336, 1104679254346, 37525681919856, 1274775209167896, 43304782313176656, 1471088177488196276, 49973690736096892016, 1697634414511896630376, 57669596280038205388752, 1959068639950002397935907

Offset: 0

Paul D. Hanna, Aug 05 2012

nonn

More generally, exp(Sum_{k>=1} A002203(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A002203(2*k)*x - x^2)^binomial(2*n,n-k).

Compare to g.f. exp(Sum_{k>=1} A002203(k) * x^k/k) = 1/(1-2*x-x^2).

			G.f.: A(x) = 1 + 16*x + 776*x^2 + 23856*x^3 + 834596*x^4 + 28135056*x^5 +...
where
log(A(x)) = 2^4*x + 6^4*x^2/2 + 14^4*x^3/3 + 34^4*x^4/4 + 82^4*x^5/5 + 198^4*x^6/6 + 478^4*x^7/7 + 1154^4*x^8/8 +...+ A002203(n)^4*x^n/n +...

Cf. A204062, A212442, A203804.

{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^4*x^k/k)+x*O(x^n)), n)}

{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n, m=2)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m, m) * prod(k=1, m, 1/(1 - (-1)^(m-k)*A002203(2*k)*x + x^2+x*O(x^n))^binomial(2*m, m-k)), n)}

G.f.: 1/((1-x)^6*(1+6*x+x^2)^4*(1-34*x+x^2)).

cp's OEIS Frontend

A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.

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A026933 Self-convolution of array T given by A008288.

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A212442 G.f.: exp( Sum_{n>=1} A002203(n)^3 * x^n/n ), where A002203 is the companion Pell numbers.

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A208034 G.f.: exp( Sum_{n>=1} 2*Pell(n)^2 * x^n/n ), where Pell(n) = A000129(n).

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A208056 G.f.: exp( Sum_{n>=1} 2*Pell(n)^(2*n) * x^n/n ), where Pell(n) = A000129(n).

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A208055 G.f.: exp( Sum_{n>=1} 2*Pell(n)^4 * x^n/n ), where Pell(n) = A000129(n).

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A215171 G.f.: exp( Sum_{n>=1} A002203(n)^4 * x^n/n ), where A002203 is the companion Pell numbers.

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A208034 G.f.: exp( Sum_{n>=1} 2Pell(n)^2 x^n/n ), where Pell(n) = A000129(n).

A208056 G.f.: exp( Sum_{n>=1} 2Pell(n)^(2n) * x^n/n ), where Pell(n) = A000129(n).

A208055 G.f.: exp( Sum_{n>=1} 2Pell(n)^4 x^n/n ), where Pell(n) = A000129(n).