A052709 -id:A052709 - cp's OEIS frontend

A039981 An example of a d-perfect sequence.

1, 1, 0, 0, 1, 2, 2, 2, 1, 0, 1, 1, 2, 0, 0, 2, 0, 1, 2, 2, 0, 0, 2, 1, 1, 1, 0, 0, 1, 1, 2, 0, 0, 2, 0, 0, 2, 0, 1, 2, 2, 1, 0, 1, 1, 2, 0, 1, 2, 2, 1, 0, 1, 2, 2, 2, 0, 0, 2, 1, 1, 1, 2, 0, 2, 2, 1, 0, 0, 1, 0, 2, 1, 1, 0, 0, 1, 2, 2, 2, 1, 0, 1, 1, 2, 0, 0, 2, 0, 0, 2, 0, 1, 2, 2, 1, 0, 1, 1, 2, 0, 1, 2, 2, 1

Offset: 1

N. J. A. Sloane

nonn

D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999. DOI: 10.1007/978-1-4471-0551-0_23

a(n) = A052709(n) mod 3. - Christian G. Bower, Jun 12 2005

More terms from Christian G. Bower, Jun 12 2005

A052726 E.g.f. (1-sqrt(1-4x-4x^2))/ (2*(1+x)).

Original entry on oeis.org

0, 1, 2, 18, 216, 3720, 81360, 2172240, 68423040, 2484639360, 102190636800, 4695453100800, 238382331264000, 13251891094041600, 800600878273996800, 52229642780899584000, 3659347096696811520000, 274040260725697449984000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 682

spec := [S,{B=Prod(Z,C),S=Union(B,Z,C),C=Prod(S,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(1-Sqrt[1-4x-4x^2])/(2(1+x)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 15 2020 *)

D-finite with recurrence: {a(1)=1, a(0)=0, a(2)=2, (-4*n^3-12*n^2-8*n)*a(n) +(-22*n-12-8*n^2)*a(n+1) +(-3*n-3)*a(n+2) +a(n+3) =0.

a(n) = n!*A052709(n). - R. J. Mathar, Oct 18 2013

A108073 Triangle in A071943 with rows reversed.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 9, 7, 3, 1, 31, 24, 12, 4, 1, 113, 89, 46, 18, 5, 1, 431, 342, 183, 76, 25, 6, 1, 1697, 1355, 741, 323, 115, 33, 7, 1, 6847, 5492, 3054, 1376, 520, 164, 42, 8, 1, 28161, 22669, 12768, 5900, 2326, 786, 224, 52, 9, 1, 117631, 94962, 54033, 25464

Offset: 0

N. J. A. Sloane, Jun 05 2005

A convolution triangle based on A052709 (with first term omitted). - Philippe Deléham, Sep 15 2005

			1; 1,1; 3,2,1; 9,7,3,1; 31,24,12,4,1; ...

Row sums yield A071356. Column 0 yields A052709.

q:=sqrt(1-4*z-4*z^2): G:=(1-q)/z/(2-t+2*z+t*q): Gserz:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gserz,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form # Emeric Deutsch, Jun 06 2005

T[n_, n_] = 1; T[n_, k_] := (k+1)*Sum[Binomial[i, n-k-i] * Binomial[k+2*i, i] / (k+i+1), {i, 1, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Vladimir Kruchinin *)

T(n,k):=if n=k then 1 else k*sum((binomial(i,n-k-i)*binomial(k+2*i-1,i))/(k+i),i,1,n-k); /* Vladimir Kruchinin, Apr 27 2015 */

G.f.: (1-q)/(z(2 - t + 2z + tq)), where q = sqrt(1 - 4z - 4z^2). - Emeric Deutsch, Jun 06 2005
T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n; T(n, k) = Sum_{j>=0} T(n-1, k-1+j) + Sum_{j>=0} T(n-1, k+1+j). - Philippe Deléham, Sep 15 2005
T(n,k) = k*Sum_{i=1..(n-k)} C(i,n-k-i)*C(k+2*i-1,i)/(k+i), n > k, T(n,n)=1. - Vladimir Kruchinin, Apr 27 2015

More terms from Emeric Deutsch, Jun 06 2005

A108075 Triangle in A071945 with rows reversed.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 9, 9, 5, 1, 31, 31, 19, 7, 1, 113, 113, 73, 33, 9, 1, 431, 431, 287, 143, 51, 11, 1, 1697, 1697, 1153, 609, 249, 73, 13, 1, 6847, 6847, 4719, 2591, 1151, 399, 99, 15, 1, 28161, 28161, 19617, 11073, 5201, 2001, 601, 129, 17, 1, 117631, 117631, 82623

Offset: 0

N. J. A. Sloane, Jun 05 2005

			Triangle begins:
   1;
   1,  1;
   3,  3,  1;
   9,  9,  5, 1;
  31, 31, 19, 7, 1;
  ...

D. Baccherini, D. Merlini and R. Sprugnoli, Level generating trees and proper Riordan arrays, Applicable Analysis and Discrete Mathematics, 2, 2008, 69-91 (see p. 88). [From _Emeric Deutsch_, Sep 21 2008]

Row sums yield A052705. Column 0 yields A052709.

q:=sqrt(1-4*z-4*z^2): G:=(1-q)/z/(1+z)/(2-t+t*q): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form - Emeric Deutsch, Jun 06 2005

G.f.: (1-q)/(z(1+z)(2-t+tq)), where q = sqrt(1 - 4z - 4z^2). - Emeric Deutsch, Jun 06 2005

T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n,k+1), T(0,0)=1. - Philippe Deléham, Nov 18 2009

More terms from Emeric Deutsch, Jun 06 2005

A128753 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDUDU's (n >= 0; 0 <= k <= n-2 for n >= 2).

Original entry on oeis.org

1, 1, 3, 9, 1, 31, 4, 1, 113, 19, 4, 1, 431, 86, 21, 4, 1, 1697, 393, 101, 23, 4, 1, 6847, 1800, 492, 116, 25, 4, 1, 28161, 8279, 2388, 596, 131, 27, 4, 1, 117631, 38218, 11603, 3032, 705, 146, 29, 4, 1, 497665, 177013, 56407, 15403, 3732, 819, 161, 31, 4, 1

Offset: 0

Emeric Deutsch, Apr 01 2007

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps.
Rows 0 and 1 have one term each; row n has n-1 terms (n >= 2).
Row sums yield A002212.

			T(4,1)=4 because we have (UDUDU)UDD, (UDUDU)UDL, U(UDUDU)DD and U(UDUDU)DL (the subwords UDUDU are shown between parentheses).
Triangle starts
    1;
    1;
    3;
    9,  1;
   31,  4,  1;
  113, 19,  4,  1;

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Cf. A002212, A052709, A026376.

C:=z->(1-sqrt(1-4*z))/2/z: G:=C(z*(1+z-t*z)/(1-t*z)): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: 1; 1; for n from 2 to 12 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form

T(n,0) = A052709(n+1).
Sum_{k=0..n-2} k*T(n,k) = A026376(n-2).
G.f.: G = G(t,z) satisfies z(1 + z - tz)G^2 - (1 - tz)G + 1 - tz = 0. G = C((1+z-tz)/(1-tz)), where C(z) = (1 - sqrt(1 - 4z))/(2z) is the Catalan function.

A306519 Expansion of 2/(1 + 2x + sqrt(1 - 4x*(1 + x))).

Original entry on oeis.org

1, 0, 2, 4, 16, 56, 216, 848, 3424, 14080, 58816, 248832, 1064064, 4591744, 19970432, 87448832, 385226240, 1705979904, 7590632448, 33916934144, 152128126976, 684702330880, 3091429158912, 13997970530304, 63550155145216, 289216809762816, 1319185060069376, 6029646893252608

Offset: 0

Ilya Gutkovskiy, Feb 21 2019

nonn

Inverse binomial transform of A001003.

Cf. A001003, A005043, A052709, A118376, A174347.

nmax = 27; CoefficientList[Series[2/(1 + 2 x + Sqrt[1 - 4 x (1 + x)]), {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[n, k] Hypergeometric2F1[1 - k, -k, 2, 2], {k, 0, n}], {n, 0, 27}]

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A001003(k).
a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 23 2019
D-finite with recurrence: (n+1)*a(n) +3*(-n+1)*a(n-1) +2*(-4*n+5)*a(n-2) +4*(-n+2)*a(n-3)=0. - R. J. Mathar, Jan 25 2020

A382885 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x) * A(x) )^3.

Original entry on oeis.org

1, 3, 18, 121, 900, 7110, 58598, 498153, 4336533, 38463732, 346368351, 3158325168, 29102914959, 270582713670, 2535191045652, 23913087584045, 226892934532149, 2164080724942155, 20737076963936828, 199542537271568802, 1927347504059464995, 18679645863925666721

Offset: 0

Seiichi Manyama, Apr 08 2025

nonn

Cf. A052709, A371576.

Cf. A365178, A371483.

a(n, r=3, s=1, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = ( 1 + x * (1+x) * A(x)^(4/3) )^3.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^3, where B(x) is the g.f. of A365178.

cp's OEIS Frontend

A039981 An example of a d-perfect sequence.

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A052726 E.g.f. (1-sqrt(1-4*x-4*x^2))/ (2*(1+x)).

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A108073 Triangle in A071943 with rows reversed.

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Maxima

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A108075 Triangle in A071945 with rows reversed.

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A128753 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDUDU's (n >= 0; 0 <= k <= n-2 for n >= 2).

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A306519 Expansion of 2/(1 + 2*x + sqrt(1 - 4*x*(1 + x))).

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A382885 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x) * A(x) )^3.

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A052726 E.g.f. (1-sqrt(1-4x-4x^2))/ (2*(1+x)).

A306519 Expansion of 2/(1 + 2x + sqrt(1 - 4x*(1 + x))).