A081182 -id:A081182 - cp's OEIS frontend

A367211 Triangular array read by rows: T(n, k) = binomial(n, k) * A000129(n - k) for 0 <= k < n.

1, 2, 2, 5, 6, 3, 12, 20, 12, 4, 29, 60, 50, 20, 5, 70, 174, 180, 100, 30, 6, 169, 490, 609, 420, 175, 42, 7, 408, 1352, 1960, 1624, 840, 280, 56, 8, 985, 3672, 6084, 5880, 3654, 1512, 420, 72, 9, 2378, 9850, 18360, 20280, 14700, 7308, 2520, 600, 90, 10

Offset: 1

Clark Kimberling, Nov 13 2023

T(n, k) are the coefficients of the polynomials p(1, x) = 1, p(2, x) = 2 + 2*x, p(n, x) = u*p(n-1, x) + v*p(n-2, x) for n >= 3, where u = p(2, x), v = 1 - 2*x - x^2.

Because (p(n, x)) is a strong divisibility sequence, for each integer k, the sequence (p(n, k)) is a strong divisibility sequence of integers.

			First nine rows:
  [n\k] 0     1     2     3     4     5    6   7  8
  [1]   1;
  [2]   2     2;
  [3]   5     6    3;
  [4]  12    20    12     4;
  [5]  29    60    50    20     5;
  [6]  70   174   180   100    30     6;
  [7] 169   490   609   420   175    42   7;
  [8] 408  1352  1960  1624   840   280   56   8;
  [9] 985  3672  6084  5880  3654  1512  420  72  9;
.
Row 4 represents the polynomial p(4,x) = 12 + 20 x + 12 x^2 + 4 x^3, so that (T(4,k)) = (12, 20, 12, 4), k = 0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A000129 (column 1, Pell numbers), A361732 (column 2), A000027 (T(n,n-1)), A007070 (row sums, p(n,1)), A077957 (alternating row sums, p(n,-1)), A081179 (p(n,2)), A077985 (p(n,-2)), A081180 (p(n,3)), A007070 (p(n,-3)), A081182 (p(n,4)), A094440, A367208, A367209, A367210.

P := proc(n) option remember; ifelse(n <= 1, n, 2*P(n - 1) + P(n - 2)) end:
T := (n, k) -> P(n - k) * binomial(n, k):
for n from 1 to 9 do [n], seq(T(n, k), k = 0..n-1) od;
# (after Werner Schulte)  Peter Luschny, Nov 24 2023

p[1, x_] := 1; p[2, x_] := 2 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
(* Or: *)
T[n_, k_] := Module[{P},
  P[m_] := P[m] = If[m <= 1, m, 2*P[m - 1] + P[m - 2]];
  P[n - k] * Binomial[n, k] ];
Table[T[n, k], {n, 1, 9}, {k, 0, n - 1}]  (* Peter Luschny, Mar 07 2025 *)

p(n, x) = u*p(n-1, x) + v*p(n-2, x) for n >= 3, where p(1, x) = 1, p(2, x) = 2 + 2*x, u = p(2, x), and v = 1 - 2*x - x^2.
p(n, x) = k*(b^n - c^n), where k = sqrt(1/8), b = x + 1 - sqrt(2), c = x + 1 + sqrt(2).
From Werner Schulte, Nov 24 2023 and Nov 25 2023: (Start)
The row polynomials p(n, x) = Sum_{k=0..n-1} T(n, k) * x^k satisfy the equation p'(n, x) = n * p(n-1, x) where p' is the first derivative of p.
T(n, k) = T(n-1, k-1) * n / k for 0 < k < n and T(n, 0) = A000129(n) for n > 0.
T(n, k) = A000129(n-k) * binomial(n, k) for 0 <= k < n.
G.f.: t / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)

New name using a formula of Werner Schulte by Peter Luschny, Mar 07 2025

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

Original entry on oeis.org

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

A081180 4th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

Original entry on oeis.org

0, 1, 8, 50, 288, 1604, 8800, 47944, 260352, 1411600, 7647872, 41420576, 224294400, 1214467136, 6575615488, 35602384000, 192760455168, 1043650265344, 5650555750400, 30593342288384, 165638957801472, 896804870374400

Offset: 0

Paul Barry, Mar 11 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
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 (8,-14).

Binomial transform of A081179.

Cf. A081182.

I:=[0, 1]; [n le 2 select I[n] else 8*Self(n-1)-14*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013

Join[{a=0,b=1},Table[c=8*b-14*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[x / (1 - 8 x + 14 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{8,-14},{0,1},30] (* Harvey P. Dale, Aug 17 2019 *)

[lucas_number1(n,8,14) for n in range(0, 22)] # Zerinvary Lajos, Apr 23 2009

a(n) = 8a(n-1) - 14a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1 - 8x + 14x^2).
a(n) = ((4 + sqrt(2))^n - (4 - sqrt(2))^n)/(2*sqrt(2)).
a(n) = Sum_{k=0..n} C(n,2k+1) 2^k*4^(n-2k-1).
If shifted once left, fourth binomial transform of A143095. - Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009, R. J. Mathar, Oct 15 2009
E.g.f.: exp(4*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

Modified the completing comment on the fourth binomial transform - R. J. Mathar, Oct 15 2009

A081183 6th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

Original entry on oeis.org

0, 1, 12, 110, 912, 7204, 55440, 420344, 3159168, 23618320, 176008128, 1309074656, 9724619520, 72186895936, 535605687552, 3972913788800, 29464372088832, 218493396246784, 1620132103941120, 12012809774902784, 89069225764835328

Offset: 0

Paul Barry, Mar 11 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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 (12,-34).

Binomial transform of A081182.

[n le 2 select n-1 else 12*Self(n-1)-34*Self(n-2): n in [1..25] ]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[x / (1 - 12 x + 34 x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12,-34},{0,1},30] (* Harvey P. Dale, Jul 31 2025 *)

[lucas_number1(n,12,34) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009

a(n) = 12*a(n-1) - 34*a(n-2) with n > 1, a(0)=0, a(1)=1.
G.f.: x/(1 - 12*x + 34*x^2).
a(n) = ((6 + sqrt(2))^n - (6 - sqrt(2))^n)/(2*sqrt(2)).
a(n) = Sum_{k=0..n} C(n,2k+1)*2^k*6^(n-2k-1).
E.g.f.: exp(6*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

A164301 a(n) = ((1+4sqrt(2))(5+sqrt(2))^n + (1-4sqrt(2))(5-sqrt(2))^n)/2.

Original entry on oeis.org

1, 13, 107, 771, 5249, 34757, 226843, 1469019, 9472801, 60940573, 391531307, 2513679891, 16131578849, 103501150997, 663985196443, 4259325491499, 27321595396801, 175251467663533, 1124117982508907, 7210396068827811, 46249247090573249, 296653361322692837

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

nonn

Binomial transform of A164300. Fifth binomial transform of A164587. Inverse binomial transform of A164598.

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

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 (10,-23).

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

Cf. A081182, A083880, A164587.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(5+r)^n+(1-4*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

LinearRecurrence[{10,-23},{1,13},20] (* Harvey P. Dale, Oct 15 2015 *)

my(x='x+O('x^50)); Vec((1+3*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Sep 13 2017

[( (1+3*x)/(1-10*x+23*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 13.
G.f.: (1+3*x)/(1-10*x+23*x^2).
E.g.f.: ( cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x) )*exp(5*x). - G. C. Greubel, Sep 13 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A083880(n) + 8*A081182(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*4^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

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

A163616 a(n) = ((1 + 3sqrt(2))(5 + sqrt(2))^n + (1 - 3sqrt(2))(5 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 11, 87, 617, 4169, 27499, 179103, 1158553, 7466161, 48014891, 308427207, 1979929577, 12705470009, 81516319819, 522937387983, 3354498523993, 21517425316321, 138020787111371, 885307088838327, 5678592784821737

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Binomial transform of A163615. Fifth binomial transform of A163864. Inverse binomial transform of A081183 without initial 0.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A163615, A163864, A081183.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(5+r)^n+(1-3*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

CoefficientList[Series[(1 + x)/(1 - 10 x + 23 x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jun 14 2014 *)
LinearRecurrence[{10, -23}, {1, 11}, 50] (* G. C. Greubel, Jul 30 2017 *)

x='x+O('x^50); Vec((1+x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Jul 30 2017

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 11.
G.f.: (1+x)/(1-10*x+23*x^2).
E.g.f.: exp(5*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017
a(n) = A081182(n)+A081182(n+1). - R. J. Mathar, Jul 01 2022

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 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.

cp's OEIS Frontend

A367211 Triangular array read by rows: T(n, k) = binomial(n, k) * A000129(n - k) for 0 <= k < n.

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

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A081180 4th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

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A081183 6th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

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A164301 a(n) = ((1+4*sqrt(2))*(5+sqrt(2))^n + (1-4*sqrt(2))*(5-sqrt(2))^n)/2.

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

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A163616 a(n) = ((1 + 3*sqrt(2))*(5 + sqrt(2))^n + (1 - 3*sqrt(2))*(5 - sqrt(2))^n)/2.

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A164301 a(n) = ((1+4sqrt(2))(5+sqrt(2))^n + (1-4sqrt(2))(5-sqrt(2))^n)/2.

A163616 a(n) = ((1 + 3sqrt(2))(5 + sqrt(2))^n + (1 - 3sqrt(2))(5 - sqrt(2))^n)/2.