A112657 -id:A112657 - cp's OEIS frontend

A089942 Inverse binomial matrix applied to A039599.

1, 0, 1, 1, 1, 1, 1, 3, 2, 1, 3, 6, 6, 3, 1, 6, 15, 15, 10, 4, 1, 15, 36, 40, 29, 15, 5, 1, 36, 91, 105, 84, 49, 21, 6, 1, 91, 232, 280, 238, 154, 76, 28, 7, 1, 232, 603, 750, 672, 468, 258, 111, 36, 8, 1, 603, 1585, 2025, 1890, 1398, 837, 405, 155, 45, 9, 1, 1585, 4213, 5500

Offset: 0

Paul Barry, Nov 16 2003

Reverse of A071947 - related to lattice paths. First column is A005043.
Triangle T(n,k), 0 <= k <= n, defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Feb 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Riordan array (f(x),x*g(x)), where f(x)is the o.g.f. of A005043 and g(x)is the o.g.f. of A001006. - Philippe Deléham, Nov 22 2009
Riordan array ((1+x-sqrt(1-2x-3x^2))/(2x(1+x)), (1-x-sqrt(1-2x-3x^2))/(2x)). Inverse of Riordan array ((1+x)/(1+x+x^2),x/(1+x+x^2)). E.g.f. of column k is exp(x)*(Bessel_I(k,2x)-Bessel_I(k+1,2x)).
Diagonal sums are A187306.
Simultaneous equations using the first n rows solve for diagonal lengths of odd N = (2n+1) regular polygons, with constants c^0, c^1, c^2, ...; where c = 1 + 2*cos( 2*Pi/N) = sin(3*Pi/N)/sin(Pi/N) = the third longest diagonal of N>5.  By way of example, take the first 4 rows relating to the 9-gon (nonagon), N=(2*4 + 1), with c = 1 + 2*cos(2*Pi/9) = 2.5320888.... The simultaneous equations are (1,0,0,0) = 1; (0,1,0,0) = c; (1,1,1,0) = c^2, (1,3,2,1) = c^3. The answers are 1, 2.532..., 2.879..., and 1.879...; the four distinct diagonal lengths of the 9-gon (nonagon) with edge = 1. - Gary W. Adamson, Sep 07 2011
Number of linearly independent irreducible representations of a given weight j in a tensor of given rank n. - Mikkel N. Schmidt, Aug 20 2025

			Triangle begins
   1,
   0,   1,
   1,   1,   1,
   1,   3,   2,   1,
   3,   6,   6,   3,   1,
   6,  15,  15,  10,   4,  1,
  15,  36,  40,  29,  15,  5,  1,
  36,  91, 105,  84,  49, 21,  6, 1,
  91, 232, 280, 238, 154, 76, 28, 7, 1
Production matrix is
  0, 1,
  1, 1, 1,
  0, 1, 1, 1,
  0, 0, 1, 1, 1,
  0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 1, 1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paul Barry and Aoife Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
J. A. R. Coope, R. F. Snider, and F. R. McCourt, Irreducible Cartesian Tensors, The Journal of Chemical Physics, vol 43, no 7, 1965.
Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
Toufik Mansour and Mark Shattuck, Enumeration of Catalan and smooth words according to capacity, Integers (2025) Vol. 25, Art. No. A5. See p. 29.
Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.
Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths, Eur. J. Comb. (2014) Vol. 39, 157-169. See Table 2.2.

Row sums give A002426 (central trinomial coefficients).

T:= (n,k) -> simplify(GegenbauerC(n-k,-n+1,-1/2)-GegenbauerC(n-k-1,-n+1,-1/2)): for n from 1 to 9 do seq(T(n,k), k=1..n) od; # Peter Luschny, May 12 2016
# Or by recurrence:
T := proc(n, k) option remember;
if n = k then 1 elif k < 0 or n < 0 or k > n then 0
elif k = 0 then T(n-1, 1) else T(n-1, k-1) + T(n-1, k) + T(n-1, k+1) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, May 25 2021

T[n_, k_] := GegenbauerC[n - k, -n + 1, -1/2] - GegenbauerC[n - k - 1, -n + 1, -1/2]; Table[T[n, k], {n,1,10}, {k,1,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

G.f.: (1+z-q)/[(1+z)(2z-t+tz+tq)], where q = sqrt(1-2z-3z^2).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A005043(m+n). - Philippe Deléham, Mar 22 2007
Sum_{k=0..n} T(n,k)*(2k+1) = 3^n. - Philippe Deléham, Mar 22 2007
Sum_{k=0..n} T(n,k)*2^k = A112657(n). - Philippe Deléham, Apr 01 2007
T(n,2k) + T(n,2k+1) = A109195(n,k). - Philippe Deléham, Nov 11 2008
T(n,k) = GegenbauerC(n-k,-n+1,-1/2) - GegenbauerC(n-k-1,-n+1,-1/2) for 1 <= k <= n. - Peter Luschny, May 12 2016

Edited by Emeric Deutsch, Mar 04 2004

A007854 Expansion of 1/(1 - 3xC(x)), where C(x) = (1 - sqrt(1 - 4x))/(2x) = g.f. for the Catalan numbers A000108.

Original entry on oeis.org

1, 3, 12, 51, 222, 978, 4338, 19323, 86310, 386250, 1730832, 7763550, 34847796, 156503064, 703149438, 3160160811, 14206181382, 63874779714, 287242041528, 1291872728826, 5810776384932, 26138647551564, 117587214581508

Offset: 0

Martin Klazar

Chains in rooted plane trees on n nodes.
The Hankel transform of the aerated sequence with g.f. 1/(1-3x^2c(x^2)) is also 3^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - Paul Barry, Jan 20 2007
Binomial transform of A112657. - Philippe Deléham, Nov 25 2007
Row sums of the Riordan matrix (1/sqrt(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A116395). - Emanuele Munarini, Apr 26 2011
Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from C(x) = 1/(1 - x*C(x)) = 1/(1 - 3*x*C(x)) (mod 2). - Peter Bala, Jul 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Martin Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions of arbitrary level, arXiv:math/0405597 [math.CO], 2004.
Index entries for sequences related to rooted trees

Cf. A000108, A000984, A076035, A076036, A067347, A116395, A126694.

CoefficientList[Series[(1+3Sqrt[1-4x])/(4-18x),{x,0,25}],x] (* Emanuele Munarini, Apr 26 2011 *)
nm = 25; t = NestList[Append[Accumulate[#], 3 Total[#]] &, {1}, nm];
Table[t[[n, n]], {n, nm}] (*similar to generating Catalan's triangle A009766*)
(* Li Han, Oct 23 2020 *)

makelist(kron_delta(n,0)+sum(binomial(2*n-k,n-k)*(k*3^k)/(2*n-k),k,1,n),n,0,12); /* Emanuele Munarini, Apr 26 2011 */

a(n) = (9*a(n-1)-3*A000108(n-2))/2 = 3*A049027(n-1) = A067336(n-1)*3/2 = A049027(n-1) + A067336(n-1) = A067347(3, n-1). - Henry Bottomley, Jan 16 2002
a(n) = Sum_{k>=0} A106566(n, k)*3^k. - Philippe Deléham, Aug 11 2005
The Hankel transform of this sequence is A000244 = [1, 3, 9, 27, 81, 243, 729, ...](powers of 3). - Philippe Deléham, Nov 26 2006
a(n) = Sum_{k = 0..n} C(2n,n-k)(2k+1)2^k/(n+k+1). - Paul Barry, Jan 20 2007
a(n) = Sum_{k = 0..n} A039599(n,k)*2^k. - Philippe Deléham, Sep 08 2007
a(n) = Sum_{k = 0..n} A116395(n,k). - Vladimir Kruchinin, Mar 09 2011
From Emanuele Munarini, Apr 26 2011 (Start)
a(n) = Sum_{k = 1..n} C(2*n-k,n-k)*(k*3^k)/(2*n-k), for n>0.
a(n) = (1/4)*(9/2)^n-3*Sum_{k=0..n} C(2*k,k)/(2k-1)*(9/2)^(n-k).
D-finite with recurrence: 2*(n+2)*a(n+2)-(17*n+22)*a(n+1)+18*(2*n+1)*a(n)=0. (End)
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = upper left term in M^n, M = the infinite square production matrix:
  3, 3, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ... (End)

More terms from Henry Bottomley, Jan 16 2002

A071947 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 3, 1, 1, 3, 6, 6, 3, 1, 4, 10, 15, 15, 6, 1, 5, 15, 29, 40, 36, 15, 1, 6, 21, 49, 84, 105, 91, 36, 1, 7, 28, 76, 154, 238, 280, 232, 91, 1, 8, 36, 111, 258, 468, 672, 750, 603, 232, 1, 9, 45, 155, 405, 837, 1398, 1890, 2025, 1585, 603, 1, 10, 55, 209, 605

Offset: 0

N. J. A. Sloane, Jun 15 2002

			Triangle begins
  1;
  1,  0;
  1,  1,  1;
  1,  2,  3,  1;
  1,  3,  6,  6,  3;
  1,  4, 10, 15, 15,  6;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Row sums give A002426 (central trinomial coefficients). Reversal of A089942.

Cf. A027907.

A071947_row := proc(n) local G, k; G := expand((1+x+x^2)^n):
seq(coeff(G,x,k) - coeff(G,x,k-1), k=0..n) end:
seq(print(A071947_row(n)), n=0..11); # Peter Luschny, Oct 01 2014

A027907[n_, k_] := Sum[Binomial[n, j]*Binomial[j, k - j], {j, 0, n}]; A005043[n_] := Sum[(-1)^k*Binomial[n, k]*Binomial[k, Floor[k/2]], {k, 0, n}]; T[n_, k_] := A027907[n, k] - A027907[n, k - 1]; T[n_, n_] := A005043[n]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Mar 02 2017 *)

G.f.: t*(1+t*z-q)/[(1+t*z)*(2*t^2*z +t*z - 1 + q)], where q = sqrt(1 -2*t*z -3*t^2*z^2).
Sum_{k, 0<=k<=n} T(n,k)*2^(n-k) = A112657(n). - Philippe Deléham, Apr 01 2007
T(n,k) = A027907(n,k) - A027907(n,k-1). T(n,n) = A005043(n). # Peter Luschny, Oct 01 2014

Edited by Emeric Deutsch, Mar 04 2004

A127501 Triangle read by rows :T(n,k)=Sum_{j, j>=0}A089942(n,j)*binomial(j,k).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 7, 10, 5, 1, 19, 31, 21, 7, 1, 51, 96, 79, 36, 9, 1, 141, 294, 282, 159, 55, 11, 1, 393, 897, 972, 645, 279, 78, 13, 1, 1107, 2727, 3273, 2475, 1269, 447, 105, 15, 1, 3139, 8272, 10835, 9136, 5369, 2254, 671, 136, 17, 1

Offset: 0

Philippe Deléham, Apr 01 2007

Riordan array (1/sqrt(1-2*x-3*x^2), (1+x-sqrt(1-2*x-3*x^2))/(2*sqrt(1-2*x-3*x^2))). - Philippe Deléham, Mar 06 2013

			Triangle begins:
  1;
  1, 1;
  3, 3, 1;
  7, 10, 5, 1;
  19, 31, 21, 7, 1;
  51, 96, 79, 36, 9, 1;
  141, 294, 282, 159, 55, 11, 1;
  393, 897, 972, 645, 279, 78, 13, 1;
  1107, 2727, 3273, 2475, 1269, 447, 105, 15, 1;
  3139, 8272, 10835, 9136, 5369, 2254, 671, 136, 17, 1; ...

T(n+1,1) = A055217(n).

Sum_{k=0..n} T(n,k)*x^k = A033999(n), A005043(n), A002426(n), A112657(n) for x = -2, -1, 0, 1 respectively.

cp's OEIS Frontend

A089942 Inverse binomial matrix applied to A039599.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A007854 Expansion of 1/(1 - 3*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

Extensions

A071947 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A127501 Triangle read by rows :T(n,k)=Sum_{j, j>=0}A089942(n,j)*binomial(j,k).

Views

Author

Keywords

Comments

Examples

Formula

A007854 Expansion of 1/(1 - 3xC(x)), where C(x) = (1 - sqrt(1 - 4x))/(2x) = g.f. for the Catalan numbers A000108.