A052705 -id:A052705 - cp's OEIS frontend

A073152 Triangle of numbers relating two simple context-free grammars (A052709 and A052705).

1, 1, 2, 3, 4, 7, 9, 12, 15, 24, 31, 40, 49, 58, 89, 113, 144, 171, 198, 229, 342, 431, 544, 637, 718, 811, 924, 1355, 1697, 2128, 2467, 2746, 3025, 3364, 3795, 5492, 6847, 8544, 9837, 10854, 11815, 12832, 14125, 15822, 22669, 28161

Offset: 0

Paul D. Hanna, Jul 29 2002

Sequence A052705 is the convolution of A052709.

			a(5,0)=a(3,3)+a(4,4)=24+89=113. a(5,3)=1*a(5,0)+1*a(4,0)+3*a(3,0)+9*a(2,0)=1*113+1*31+3*9+9*3=198. Rows: {1}; {1,2}; {3,4,7}; {9,12,15,24}; {31,40,49,58,89}; {113,144,171,198,229,342}; {431,544,637,718,811,924,1355}; {1697,2128,2467,2746,3025,3364,3795,5492}

D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math. 49 (1997) 301

Cf. A052709, A052705.

Triangle {a(n, k), n >= 0, 0<=k<=n} defined by: a(0, 0)=1, a(n, 0)=A052709(n+1), a(n, n)=A052705(n+2), a(n, 0)=a(n-1, n-1) + a(n-2, n-2), a(n, k)=sum{j=0..k} A052709(j+1) * a(n-j, 0).

A371576 G.f. satisfies A(x) = ( 1 + xA(x)^(3/2) (1 + x) )^2.

Original entry on oeis.org

1, 2, 9, 44, 240, 1390, 8404, 52426, 334964, 2180928, 14418123, 96525656, 653077411, 4458529390, 30674865164, 212472058410, 1480446579602, 10369560147798, 72972217926122, 515674254743332, 3657933383804959, 26036659997517572, 185905008055923918

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Column k=2 of A378323.
Cf. A001629, A052705, A371577, A371578.
Cf. A270386, A364475.

a(n, r=2, s=1, t=3, 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));

a(n) = 2 * Sum_{k=0..n} binomial(3*k+2,k) * binomial(k,n-k)/(3*k+2).

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A364475.

A371578 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) (1 + x) )^2.

Original entry on oeis.org

1, 2, 13, 102, 916, 8880, 90607, 958794, 10426089, 115798342, 1308035135, 14980661482, 173553196140, 2030265152576, 23948922940698, 284543368174220, 3402103050539715, 40903437537402792, 494215527894112099, 5997782678374854902, 73078635875447981850

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Cf. A001629, A052705, A371576, A371577.

Cf. A365184, A371580.

a(n, r=2, s=1, t=5, 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));

a(n) = 2 * Sum_{k=0..n} binomial(5*k+2,k) * binomial(k,n-k)/(5*k+2).

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A365184.

A371577 G.f. satisfies A(x) = ( 1 + xA(x)^2 (1 + x) )^2.

Original entry on oeis.org

1, 2, 11, 70, 505, 3910, 31772, 267280, 2307982, 20339946, 182207333, 1654250474, 15187764411, 140767293560, 1315349040350, 12377806027892, 117200381305538, 1115791797318548, 10674418686087377, 102563189093302366, 989321056200478417

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Cf. A001629, A052705, A371576, A371578.

Cf. A365178.

a(n, r=2, 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));

a(n) = Sum_{k=0..n} binomial(4*k+2,k) * binomial(k,n-k)/(2*k+1).

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A365178.

A110681 A convolution triangle of numbers based on A071356.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 20, 16, 6, 1, 72, 64, 30, 8, 1, 272, 260, 140, 48, 10, 1, 1064, 1072, 636, 256, 70, 12, 1, 4272, 4480, 2856, 1288, 420, 96, 14, 1, 17504, 18944, 12768, 6272, 2320, 640, 126, 16, 1, 72896, 80928, 57024, 29952, 12192, 3852, 924, 160, 18, 1

Offset: 0

Philippe Deléham, Sep 14 2005

			Triangle starts:
   1;
   2,  1;
   6,  4,  1;
  20, 16,  6,  1;
  72, 64, 30,  8,  1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11324 (rows 0 <= n <= 150).
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), p. 32-33.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

Cf. A071356 (1st column), A071357 (2nd column).

Cf. A052705 (diagonal sums), A001003 (row sums).

T[n_, k_] := T[n, k] = Which[n == k == 0, 1, n == 0, 0, k == 0, 0, k > n, 0, True, T[n - 1, k - 1] + 2 T[n - 1, k] + 2 T[n - 1, k + 1]]; Table[T[n, k], {n, 0, 10}, {k, n}] // Flatten (* Michael De Vlieger, Nov 05 2017 *)

T(0, 0) = 1; T(n, k) = 0 if k<0 or if k>n; T(n, k) = T(n-1, k-1) + 2*T(n-1, k) + 2*T(n-1, k+1).
Sum_{k, k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A071356(m+n).
Sum_{k, k>=0} T(n, k)*(2^(k+1) - 1) = 5^n.
Sum_{k, k>=0} (-1)^(n-k)*T(n, k)*(2^(k+1) - 1) = 1.

A052728 A simple context-free grammar in a labeled universe.

Original entry on oeis.org

0, 0, 2, 12, 168, 2880, 64080, 1723680, 54633600, 1992936960, 82261267200, 3790579161600, 192895381324800, 10744251136819200, 650181362358528000, 42476922345521664000, 2979716339168464896000, 223385082959833546752000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 684

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

E.g.f.: (1/2)/(x^2+1+2*x)*(1-2*x-2*x^2-(1-4*x-4*x^2)^(1/2))
Recurrence: {a(1)=0, a(2)=2, a(3)=12, (-4*n^3-16*n^2-20*n-8)*a(n) +(-8*n^2-26*n-20)*a(n+1) +(-2-3*n)*a(n+2) +a(n+3) =0.
a(n) = n!*A052705(n). - R. J. Mathar, Oct 18 2013

A052744 E.g.f. x(1-2x-2x^2-sqrt(1-4x-4x^2))/ (2(1+x)^2).

Original entry on oeis.org

0, 0, 0, 6, 48, 840, 17280, 448560, 13789440, 491702400, 19929369600, 904873939200, 45486949939200, 2507639957222400, 150419515915468800, 9752720435377920000, 679630757528346624000, 50655177765863903232000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 700

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

D-finite with recurrence: a(1)=0, a(2)=0, a(3)=6, (-4*n^4-24*n-24*n^3-44*n^2)*a(n) +(-8*n^3-42*n^2-12-58*n)*a(n+1) +(-3*n^2-8*n+3)*a(n+2) +(n+2)*a(n+3)=0, a(4)=48, a(5)=840, a(6)=17280.
a(n) ~ sqrt(58-41*sqrt(2))*(1+sqrt(2))^(n-1)*2^n*n^(n-1)*exp(-n). - Vaclav Kotesovec, Aug 18 2013
a(n)= n!*A052705(n-1). - R. J. Mathar, Oct 26 2013

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

A162303 Product matrix [C(k,n-k)]*A001263.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 3, 12, 8, 1, 5, 31, 39, 13, 1, 8, 73, 148, 93, 19, 1, 13, 162, 481, 486, 186, 26, 1, 21, 344, 1406, 2080, 1274, 332, 34, 1, 34, 707, 3803, 7741, 6920, 2873, 547, 43, 1

Offset: 0

Paul Barry, Jun 30 2009

First column is A000045. Row sums are A052705(n+2).

			Triangle begins
1,
1, 1,
2, 4, 1,
3, 12, 8, 1,
5, 31, 39, 13, 1,
8, 73, 148, 93, 19, 1,
13, 162, 481, 486, 186, 26, 1,
21, 344, 1406, 2080, 1274, 332, 34, 1

Number triangle T(n,k)=sum{j=0..n, C(j,n-j)*C(j+1,k)*C(j+1,k+1)/(j+1)}.

cp's OEIS Frontend

A073152 Triangle of numbers relating two simple context-free grammars (A052709 and A052705).

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A371576 G.f. satisfies A(x) = ( 1 + x*A(x)^(3/2) * (1 + x) )^2.

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A371578 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) * (1 + x) )^2.

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A371577 G.f. satisfies A(x) = ( 1 + x*A(x)^2 * (1 + x) )^2.

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PARI

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A110681 A convolution triangle of numbers based on A071356.

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Mathematica

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A052728 A simple context-free grammar in a labeled universe.

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A052744 E.g.f. x*(1-2*x-2*x^2-sqrt(1-4*x-4*x^2))/ (2*(1+x)^2).

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Maple

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A108075 Triangle in A071945 with rows reversed.

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Maple

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A162303 Product matrix [C(k,n-k)]*A001263.

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Formula

A371576 G.f. satisfies A(x) = ( 1 + xA(x)^(3/2) (1 + x) )^2.

A371578 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) (1 + x) )^2.

A371577 G.f. satisfies A(x) = ( 1 + xA(x)^2 (1 + x) )^2.

A052744 E.g.f. x(1-2x-2x^2-sqrt(1-4x-4x^2))/ (2(1+x)^2).