A103209 -id:A103209 - cp's OEIS frontend

A107703 Sums of antidiagonals of number array A103209.

1, 2, 4, 11, 43, 218, 1336, 9557, 78053, 714578, 7223548, 79704015, 951626175, 12210506762, 167413540912, 2440529176297, 37665520151241, 613124748803618, 10492684074110772, 188241358671950227, 3531254422923432083

Offset: 0

Paul Barry, May 21 2005

Row sums of A107702.

a(n)=sum{k=0..n, sum{j=0..n-k, C(n-k+j, 2j)*k^j*C(k)}}, C(n) given by A000108.

A107704 Diagonal sums of A103209, viewed as number triangle.

Original entry on oeis.org

1, 1, 2, 3, 8, 26, 107, 492, 2481, 13599, 81288, 531343, 3790344, 29279668, 242278645, 2125160800, 19608039385, 189437263949, 1912477987102, 20161911603747, 221869317899264, 2546362514225134, 30430660439311103

Offset: 0

Paul Barry, May 21 2005

a(n)=sum{k=0..floor(n/2), sum{j=0..n-2k, C(n-2k+j, 2j)*k^j*C(k)}}, C(n) given by A000108.

A103210 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)2^i*3^(n-i), a(0)=1.

Original entry on oeis.org

1, 3, 15, 93, 645, 4791, 37275, 299865, 2474025, 20819307, 178003815, 1541918901, 13503125805, 119352115551, 1063366539315, 9539785668657, 86104685123025, 781343125570515, 7124072211203775, 65233526296899981, 599633539433039445, 5531156299278726663

Offset: 0

Ralf Stephan, Jan 27 2005

nonn

The Hankel transform of this sequence is 6^C(n+1,2). - Philippe Deléham, Oct 28 2007
The Hankel transform of the sequence starting 1, 1, 3, 15, ... is A081955. - Paul Barry, Dec 09 2008
Number of Schroeder paths from (0,0) to (0,2n) allowing two colors for the down steps (or alternatively for the rise steps). - Paul Barry, Feb 01 2009
Essentially, reversion of x*(1-2*x)/(1+x). - Paul Barry, Apr 28 2009
a(n) is also the number of infix expressions with n variables and operators +, - and * (or +, * and /) such that there are no redundant parentheses. - Vjeran Crnjak, Apr 25 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Joseph Abate and Ward Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5. b_n(2).
Eyal Ackerman, Gill Barequet, Ron Y. Pinter, and Dan Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett. (2006) Vol. 98, No. 4, 162-167.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016, eq. (1.13), a=3, b=2.
Robert Dickau, 3D Guillotine Partitions, figures for 3D slices.
Samuele Giraudo, Operads from posets and Koszul duality, arXiv preprint arXiv:1504.04529 [math.CO], 2015.
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Third column of array A103209.

Cf. A000108, A060693, A081955, A107841.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!((1-x-Sqrt(x^2-10*x+1))/(4*x))); // G. C. Greubel, Feb 10 2018

A103210 := proc(n)
    if n = 0 then
        1;
    else
        add(binomial(n,i)*binomial(n,i+1)*2^i*3^(n-i),i=0..n-1)/n ;
    end if;
end proc: # R. J. Mathar, Feb 10 2015
A103210_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 3*a[w-1] + 2*add(a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list) end: A103210_list(21); # Peter Luschny, Feb 29 2016

CoefficientList[Series[(1-x-Sqrt[x^2-10*x+1])/(4*x), {x, 0, 25}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
A103210[n_]:= Hypergeometric2F1[-n, n+1, 2, -2]; Table[A103210[n], {n, 0, 25}] (* Peter Luschny, Jan 07 2018 *)

my(x='x+O('x^25)); Vec((1-x-sqrt(x^2-10*x+1))/(4*x)) \\ G. C. Greubel, Feb 10 2018

[1]+[(3^n/n)*sum( binomial(n,j)*binomial(n,j+1)*(2/3)^j for j in (0..n-1)) for n in (1..25)] # G. C. Greubel, Jun 08 2020

G.f.: (1 - z - sqrt(1 -10*z +z^2))/(4*z).
a(n) = Sum_{k=0..n} C(n+k, 2k) * 2^k * C(k), C(n) given by A000108. - Paul Barry, May 21 2005
a(n) = Sum_{k=0..n} A060693(n,k)*2^(n-k). - Philippe Deléham, Apr 02 2007
a(0) = 1, a(n) = a(n-1) + 2*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
a(n) = (3/2)*A107841(n) for n > 0. - Philippe Deléham, Oct 28 2007
G.f.: 1/(1-x-2*x/(1-x-2*x/(1-x-2*x/(1-.... (continued fraction). - Paul Barry, Feb 01 2009
G.f.: 1/(1-3*x-6*x^2/(1-5*x-6*x^2/(1-5*x-6*x^2/(1-... (continued fraction). - Paul Barry, Apr 28 2009
G.f.: 1/(1-3*x/(1-2*x/(1-3*x/(1-2*x/(1-3*x/(1-... (continued fraction). - Paul Barry, May 14 2009
a(n) = Hypergeometric2F1(-n,n+1;2;-2) = Sum_{k=0..n} C(n+k,k) * C(n,k) * 2^k/(k+1). - Paul Barry, Feb 08 2011
G.f.: A(x) = (1-x-(x^2-10*x+1)^(1/2))/(4*x) = 1/(G(0)-x); G(k)= 1 + x - 3*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012
D-finite with recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(12+5*sqrt(6))*(5+2*sqrt(6))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A103211 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)3^i*4^(n-i), a(0)=1.

Original entry on oeis.org

1, 4, 28, 244, 2380, 24868, 272188, 3080596, 35758828, 423373636, 5092965724, 62071299892, 764811509644, 9511373563492, 119231457692284, 1505021128450516, 19112961439180588, 244028820862442116, 3130592301487969948, 40333745806536135028, 521655330655122923980

Offset: 0

Ralf Stephan, Jan 27 2005

nonn

The Hankel transform of this sequence is 12^C(n+1,2). - Philippe Deléham, Oct 28 2007
The sequence 1, 1, 4, 28, ... has a(n) = 0^n + Sum_{k=0..n-1} C(n+k-1, 2*k)*C(k)*3^k and Hankel transform 3^C(n+1, 2)*4^C(n, 2). - Paul Barry, Dec 09 2008
Number of Dyck n-paths with two colors of up (U,u) and two colors of down (D,d) avoiding DU. - David Scambler, Jun 24 2013

			G.f. = 1 + 4*x + 28*x^2 + 244*x^3 + 2380*x^4 + 24868*x^5 + ... _Michael Somos_, Mar 15 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, b_n(3).
E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett. 98 (4) (2006) 162-167
Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], (2016), eq. (1.13), a=4, b=3.
Samuele Giraudo, Operads from posets and Koszul duality, arXiv preprint arXiv:1504.04529 [math.CO], 2015.
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.
Djamila Oudrar, Sur l'énumération de structures discrètes, une approche par la théorie des relations, Thesis (in French), arXiv:1604.05839 [math.CO], 2016.

Fourth column of array A103209.

Cf. A131763.

a:=n->(1/n)*Sum([0..n-1],i->Binomial(n,i)*Binomial(n,i+1)*
3^i*4^(n-i));;
A103211:=Concatenation([1],List([1..20],n->a(n))); # Muniru A Asiru, Feb 11 2018

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-14*x+1))/(6*x))) // G. C. Greubel, Feb 10 2018

A103211_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1] + 3*add(a[j]*a[w-j-1], j=0..w-1) od;
convert(a, list) end: A103211_list(20); # Peter Luschny, Feb 29 2016

CoefficientList[Series[(1-x-Sqrt[x^2-14*x+1])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
a[n_] := Sum[(-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, 4], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Jan 08 2018 *)
a[ n_] := If[n < 0, a[-1-n], SeriesCoefficient[2/(1 - x + Sqrt[1 - 14*x + x^2]), {x, 0, n}]]; (* Michael Somos, Mar 15 2024 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-14*x+1))/(6*x)) \\ G. C. Greubel, Feb 10 2018

{a(n) = if(n<0, a(-1-n), polcoeff(2/(1 - x + sqrt(1 - 14*x + x^2 + x*O(x^n))), n))}; /* Michael Somos, Mar 15 2024 */

G.f.: (1-z-sqrt(z^2-14*z+1))/(6*z).
a(n) = Sum_{k=0..n} C(n+k,2k)*3^k*C(k), C(n) given by A000108. - Paul Barry, May 21 2005
a(n) = Sum_{k=0..n} A060693(n,k)*3^(n-k). - Philippe Deléham, Apr 02 2007
a(0)=1, a(n) = a(n-1) + 3*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
G.f.: 1/(1-x-3*x/(1-x-3*x/(1-x-3*x/(1-x-3*x/(1-... (continued fraction). - Paul Barry, Nov 07 2009
D-finite with recurrence: (n+1)*a(n) = 7*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(24+14*sqrt(3))*(7+4*sqrt(3))^n/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
a(n) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k)*hypergeom([k - n, n + 1], [k + 2], 4). - Peter Luschny, Jan 08 2018
G.f. A(x) satisfies: A(x) = (1 + 3*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020
From Michael Somos, Mar 15 2024: (Start)
a(n) = (4/3)*A131763(n) for n>0.
Given g.f. A(x) and y = -x*A(-x^2), then 3*y-1/y = x+1/x.
If a(n) := a(-1-n) for n<0, then 0 = a(n)*(+a(n+1) -35*a(n+2) +4*a(n+3)) +a(n+1)*(+7*a(n+1) +194*a(n+2) -35*a(n+3)) +a(n+2)*(+7*a(n+2) +a(n+3)) for all n in Z. (End)

A133305 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)4^i5^(n-i), a(0) = 1.

Original entry on oeis.org

1, 5, 45, 505, 6345, 85405, 1204245, 17558705, 262577745, 4005148405, 62070886845, 974612606505, 15471084667545, 247876665109005, 4003225107031845, 65101209768055905, 1065128963164067745, 17520376884067071205, 289572455530026439245, 4806489064223483202905

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Fifth column of array A103209.

The Hankel transform of this sequence is 20^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..795
Samuele Giraudo, Operads from posets and Koszul duality, arXiv:1504.04529 [math.CO], 2015-2016.

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-18*x+1))/(8*x))) // G. C. Greubel, Feb 10 2018

a[n_] := Hypergeometric2F1[-n, n + 1, 2, -4];
Table[a[n], {n, 0, 16}] (* Peter Luschny, Jan 08 2018 *)
CoefficientList[Series[(1-x-Sqrt[x^2-18*x+1])/(8*x), {x, 0, 50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-18*x+1))/(8*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-18*z+1))/(8*z).
a(n) = Sum_{k=0..n} A088617(n,k)*4^k.
a(n) = Sum_{k=0..n} A060693(n,k)*4^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)*4^k*C(k), C(n) given by A000108.
a(0) = 1, a(n) = a(n-1) + 4*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 9*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
G.f.: 1/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) = hypergeom([-n, n + 1], [2], -4). - Peter Luschny, Jan 08 2018
a(n) ~ 5^(1/4) * phi^(6*n + 3) / (2^(5/2) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 21 2021

A133306 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)5^i6^(n-i), a(0)=1.

Original entry on oeis.org

1, 6, 66, 906, 13926, 229326, 3956106, 70572066, 1291183806, 24095736726, 456879955026, 8776867331706, 170459895028566, 3341423256586206, 66023812564384026, 1313634856606430226, 26295597219228901806, 529199848207277494566, 10701116421278640683106, 217317899302044152030826

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Sixth column of array A103209.

The Hankel transform of this sequence is 30^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..745

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-22*x+1))/(10*x))) // G. C. Greubel, Feb 10 2018

CoefficientList[Series[(1-x-Sqrt[x^2-22*x+1])/(10*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-22*x+1))/(10*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-22*z+1))/(10*z).
a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*5^k.
a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*5^(n-k).
a(n) = Sum_{k, 0<=k<=n} C(n+k, 2*k) 5^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 5*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 11*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
G.f.: 1/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) ~ 3^(1/4) * (11 + 2*sqrt(30))^(n + 1/2) / (10^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 29 2021

A133307 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)6^i7^(n-i), a(0)=1.

Original entry on oeis.org

1, 7, 91, 1477, 26845, 522739, 10663471, 224939113, 4866571801, 107393779423, 2407939176643, 54700070934061, 1256249370578293, 29119953189833611, 680401905145643863, 16008309928027493713, 378930780842531820721, 9017843351806985482423, 215634517504141993966891

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Seventh column of array A103209.

The Hankel transform of this sequence is 42^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..700

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-26*x+1))/(12*x))) // G. C. Greubel, Feb 10 2018

a := n -> hypergeom([-n, n+1], [2], -6);
seq(round(evalf(a(n),32)),n=0..16); # Peter Luschny, May 23 2014

CoefficientList[Series[(1-x-Sqrt[x^2-26*x+1])/(12*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-26*x+1))/(12*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-26*z+1))/(12*z).
a(n) = Sum_{k=0..n} A088617(n,k)*6^k .
a(n) = Sum_{k=0..n} A060693(n,k)*6^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)6^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 6*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 13*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
a(n) = hypergeom([-n, n+1], [2], -6). # Peter Luschny, May 23 2014
G.f.: 1/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) ~ 42^(1/4) * (13 + 2*sqrt(42))^(n + 1/2) / (12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 29 2021

A133308 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)7^i8^(n-i), a(0)=1.

Original entry on oeis.org

1, 8, 120, 2248, 47160, 1059976, 24958200, 607693640, 15175702200, 386555020552, 10004252294520, 262321706465736, 6953918939056440, 186059575955360136, 5018045415643478520, 136276936332343342152, 3723442515218861494200, 102281105054908404972040

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Eighth column of array A103209.

The Hankel transform of this sequence is 56^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..675

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-30*x+1))/(14*x))) // G. C. Greubel, Feb 10 2018

a := n -> hypergeom([-n, n+1], [2], -7);
seq(round(evalf(a(n), 32)), n=0..15); # Peter Luschny, May 23 2014

CoefficientList[Series[(1-x-Sqrt[x^2-30*x+1])/(14*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-30*x+1))/(14*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-30*z+1))/(14*z).
a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*7^k.
a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*7^(n-k).
a(n) = Sum_{k, 0<=k<=n} C(n+k, 2k)7^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 7*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 15*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
a(n) = hypergeom([-n, n+1], [2], -7). - Peter Luschny, May 23 2014
G.f.: 1/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

A133309 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)8^i9^(n-i), a(0)=1.

Original entry on oeis.org

1, 9, 153, 3249, 77265, 1968633, 52546473, 1450365921, 41058670113, 1185580310121, 34783088255289, 1033907690362257, 31070005849929969, 942384250116160857, 28812102048874578249, 887007207177728561601, 27473495809057571051073, 855518113376312857290441

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Ninth column of array A103209.

The Hankel transform of this sequence is 72^C(n+1,2). - Philippe Deléham, Oct 29 2007

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

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!( (1-x-Sqrt(x^2-34*x+1))/16 )); // G. C. Greubel, Feb 10 2018

Rest@ CoefficientList[ Series[(1-x-Sqrt[x^2-34*x+1])/16, {x, 0, 18}], x] (* Robert G. Wilson v, Oct 19 2007 *)
Table[-((3 I LegendreP[n, -1, 2, 17])/(2 Sqrt[2])), {n, 0, 20}] (* Vaclav Kotesovec, Aug 13 2013 *)

my(x='x+O('x^30)); Vec((1-x-sqrt(x^2-34*x+1))/16) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-34*z+1))/16.
a(n) = Sum_{k=0..n} A088617(n,k)*8^k.
a(n) = Sum_{k=0..n} A060693(n,k)*8^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)8^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 8*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
a(n) ~ sqrt(144+102*sqrt(2))*(17+12*sqrt(2))^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013
Recurrence: (n+1)*a(n) = 17*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Aug 13 2013
G.f.: 1/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

More terms from Robert G. Wilson v, Oct 19 2007

A292798 a(n) = [x^n] 1/(1 - x - nx/(1 - x - nx/(1 - x - nx/(1 - x - nx/(1 - x - n*x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 2, 15, 244, 6345, 229326, 10663471, 607693640, 41058670113, 3210853971610, 285387481699551, 28423216247375676, 3136023698489382025, 379743303818657805222, 50074394496591697023135, 7143088376895580682492176, 1096075604718147681983312001, 180030794404631168482202007090

Offset: 0

Ilya Gutkovskiy, Sep 23 2017

nonn

Main diagonal of A103209 (with offset 0).

Cf. A006318.

Table[SeriesCoefficient[1/(1 - x + ContinuedFractionK[-n x, 1 - x, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]
Table[SeriesCoefficient[(1 - x + Sqrt[1 - 2 (2 n + 1) x + x^2])/(1 - 2 (n + 1) x + x^2 - (x - 1) Sqrt[1 - 2 (2 n + 1) x + x^2]), {x, 0, n}], {n, 0, 17}]
Table[Hypergeometric2F1[-n, n + 1, 2, -n], {n, 0, 17}]

{a(n) = polcoeff( (1+x)^(n+1) / (1 - n*x +x*O(x^n) )^(n+1), n) / (n+1)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, May 07 2018

a(n) ~ exp(1/2) * 2^(2*n) * n^(n-3/2) / sqrt(Pi). - Vaclav Kotesovec, Sep 24 2017
a(n) = (1/(n+1)) * [x^n] (1+x)^(n+1) / (1 - n*x)^(n+1). - Paul D. Hanna, May 07 2018
From Fabian Pereyra, Sep 02 2024: (Start)
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*n^k/(k+1).
a(n) = [x^n] 2/(1 - x + sqrt(1 - 2*x*(1 + 2*n) + x^2)). (End)

cp's OEIS Frontend

A107703 Sums of antidiagonals of number array A103209.

Views

Author

Keywords

Comments

Formula

A107704 Diagonal sums of A103209, viewed as number triangle.

Views

Author

Keywords

Formula

A103210 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*2^i*3^(n-i), a(0)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A103211 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*3^i*4^(n-i), a(0)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Formula

A133305 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*4^i*5^(n-i), a(0) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A133306 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*5^i*6^(n-i), a(0)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A133307 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*6^i*7^(n-i), a(0)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A103210 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)2^i*3^(n-i), a(0)=1.

A103211 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)3^i*4^(n-i), a(0)=1.

A133305 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)4^i5^(n-i), a(0) = 1.

A133306 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)5^i6^(n-i), a(0)=1.

A133307 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)6^i7^(n-i), a(0)=1.

A133308 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)7^i8^(n-i), a(0)=1.

A133309 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)8^i9^(n-i), a(0)=1.

A292798 a(n) = [x^n] 1/(1 - x - nx/(1 - x - nx/(1 - x - nx/(1 - x - nx/(1 - x - n*x/(1 - ...)))))), a continued fraction.