A081184 -id:A081184 - cp's OEIS frontend

A081179 3rd binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

0, 1, 6, 29, 132, 589, 2610, 11537, 50952, 224953, 993054, 4383653, 19350540, 85417669, 377052234, 1664389721, 7346972688, 32431108081, 143157839670, 631929281453, 2789470811028, 12313319895997, 54353623698786

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of 0, 1, 4, 14, 48, ... (A007070 with offset 1) and second binomial transform of A000129. - R. J. Mathar, Dec 10 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Sergio Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A081180, A081182, A081183, A081184, A081185, A153593.

I:=[0, 1]; [n le 2 select I[n] else 6*Self(n-1)-7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013

f:= gfun:-rectoproc({a(n) = 6*a(n-1)-7*a(n-2), a(0)=0, a(1)=1},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 15 2016

CoefficientList[Series[x/(1-6 x +7 x^2), {x,0,30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{6,-7}, {0,1}, 41] (* G. C. Greubel, Jan 14 2024 *)

[lucas_number1(n,6,7) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009

a(n) = 6*a(n-1) - 7*a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1-6*x+7*x^2).
a(n) = ((3+sqrt(2))^n - (3-sqrt(2))^n)/(2*sqrt(2)). [Corrected by Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008]
a(n) = 3^(n-1) Sum_{i>=0} binomial(n, 2i+1) * (2/9)^i. - Sergio Falcon, Mar 15 2016
a(n) = 2^(-1/2)*7^(n/2)*sinh(n*arcsinh(sqrt(2/7))). - Robert Israel, Mar 15 2016
E.g.f.: exp(3*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017
a(n) = 7^((n-1)/2)*ChebyshevU(n-1, 3/sqrt(7)). - G. C. Greubel, Jan 14 2024

A081185 8th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

Original entry on oeis.org

0, 1, 16, 194, 2112, 21764, 217280, 2127112, 20562432, 197117968, 1879016704, 17842953248, 168988216320, 1597548359744, 15083504344064, 142288071200896, 1341431869882368, 12641049503662336, 119088016125890560

Offset: 0

Paul Barry, Mar 11 2003

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (16,-62).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), this sequence (m=7), A164600 (m=8).
Binomial transform of A081184.
Cf. A081185, A147959.

[n le 2 select n-1 else 16*Self(n-1)-62*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 07 2013

m:=30; S:=series( x/(1-16*x+62*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 12 2021

Join[{a=0,b=1},Table[c=16*b-62*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
CoefficientList[Series[x/(1-16x+62x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{16,-62},{0,1},30] (* Harvey P. Dale, Sep 24 2013 *)

[( x/(1-16*x+62*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = 16*a(n-1) - 62*a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1 - 16*x + 62*x^2).
a(n) = ((8 + sqrt(2))^n - (8 - sqrt(2))^n)/(2*sqrt(2)).
a(n) = Sum_{k=0..n} C(n,2*k+1) * 2^k * 7^(n-2*k-1).
E.g.f.: exp(8*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A147959(n) + 8*A081185(n).
a(n) = (1/2)*Sum_{k=0..n-1} binomial(n-1,k)*7^(n-k-1)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A164599 a(n) = 14a(n-1) - 47a(n-2), for n > 1, with a(0) = 1, a(1) = 15.

Original entry on oeis.org

1, 15, 163, 1577, 14417, 127719, 1110467, 9543745, 81420481, 691330719, 5851867459, 49433600633, 417032638289, 3515077706295, 29610553888547, 249339102243793, 2099051398651393, 17667781775661231, 148693529122641763

Offset: 0

Klaus Brockhaus, Aug 17 2009

nonn

Binomial transform of A164598. Seventh binomial transform of A164587. Inverse binomial transform of A081185 without initial term 0.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 -2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 -2)*x^2) and have a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 11 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (14,-47).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), this sequence (m=6), A081185 (m=7), A164600 (m=8).

Cf. A002203, A081184, A081185, A147958, A164587.

[ n le 2 select 14*n-13 else 14*Self(n-1)-47*Self(n-2): n in [1..30] ];

m:=30; S:=series( (1+x)/(1-14*x+47*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 11 2021

LinearRecurrence[{14,-47}, {1,15}, 30] (* G. C. Greubel, Aug 11 2017 *)

my(x='x+O('x^30)); Vec((1+x)/(1-14*x+47*x^2)) \\ G. C. Greubel, Aug 11 2017

def A164599_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/(1-14*x+47*x^2) ).list()
A164599_list(30) # G. C. Greubel, Mar 11 2021

a(n) = ((1+4*sqrt(2))*(7+sqrt(2))^n + (1-4*sqrt(2))*(7-sqrt(2))^n)/2.
G.f.: (1+x)/(1-14*x+47*x^2).
E.g.f.: exp(7*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
From G. C. Greubel, Mar 11 2021: (Start)
a(n) = A147958(n) + 8*A081184(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*6^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A153593 a(n) = ((9 + sqrt(2))^n - (9 - sqrt(2))^n)/(2*sqrt(2)).

Original entry on oeis.org

1, 18, 245, 2988, 34429, 383670, 4186169, 45041112, 480032665, 5082340122, 53559541661, 562566880260, 5895000053461, 61667217421758, 644304909368225, 6725778192309168, 70163919621475249, 731614075994130210

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

nonn

Preceded by zero, this is the eighth binomial transform of the Pell sequence A000129. - Sergio Falcon, Sep 21 2009; edited by Klaus Brockhaus, Oct 11 2009
Eighth binomial transform of A048697.
First differences are in A164600.
lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(2) = 10.4142135623....

Vincenzo Librandi, Table of n, a(n) for n = 1..100
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (18,-79).

Cf. A000129 (Pell numbers), A007070, A081185, A081184, A081183, A081182, A081180, A081179. - Sergio Falcon, Sep 21 2009

Cf. A002193 (decimal expansion of sqrt(2)), A048697, A164600.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008

Join[{a=1,b=18},Table[c=18*b-79*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{18,-79},{1,18},25] (* G. C. Greubel, Aug 22 2016 *)

a(n) = 18*a(n-1) - 79*a(n-2) for n>1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
G.f.: x/(1 - 18*x + 79*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009
a(n) = Sum[Binomial[n - 1 - i, i] (-1)^i * 18^(n - 1 - 2 i) * 79^i, {i, 0, Floor[(n - 1)/2]}]. - Sergio Falcon, Sep 21 2009
E.g.f.: exp(9*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

A171700 Triangle T : T(n,k)= A007318(n,k)*a(n-k) with a(0)=0 and a(n)= A077957(n-1) for n>0.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 2, 0, 3, 0, 0, 8, 0, 4, 0, 4, 0, 20, 0, 5, 0, 0, 24, 0, 40, 0, 6, 0, 8, 0, 84, 0, 70, 0, 7, 0, 0, 64, 0, 224, 0, 112, 0, 8, 0, 16, 0, 288, 0, 504, 0, 168, 0, 9, 0, 0, 160, 0, 960, 0, 1008, 0, 240, 0, 10, 0

Offset: 0

Philippe Deléham, Dec 15 2009

Diagonal sums : A001353(n+1) alternating with zeros.

			Triangle begins : 0 ; 1,0 ; 0,2,0 ; 2,0,3,0 ; 0,8,0,4,0 ; 4,0,20,0,5,0 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n-1), A000129(n), A007070(n-1), A081179(n), A081180(n), A081182(n), A081183(n), A081184(n), A081185(n), A153593(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.

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A081179 3rd binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

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A081185 8th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

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A164599 a(n) = 14*a(n-1) - 47*a(n-2), for n > 1, with a(0) = 1, a(1) = 15.

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A153593 a(n) = ((9 + sqrt(2))^n - (9 - sqrt(2))^n)/(2*sqrt(2)).

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A171700 Triangle T : T(n,k)= A007318(n,k)*a(n-k) with a(0)=0 and a(n)= A077957(n-1) for n>0.

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A164599 a(n) = 14a(n-1) - 47a(n-2), for n > 1, with a(0) = 1, a(1) = 15.