A172991 -id:A172991 - cp's OEIS frontend

A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 15, 10, 1, 1, 9, 28, 35, 15, 1, 1, 11, 45, 84, 70, 21, 1, 1, 13, 66, 165, 210, 126, 28, 1, 1, 15, 91, 286, 495, 462, 210, 36, 1, 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1

Offset: 0

Clark Kimberling

Row sums are odd-indexed Fibonacci numbers.
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k double rises. Mirror image of A085478. - Emeric Deutsch, May 31 2004
Diagonal sums are A052535. - Paul Barry, Jan 21 2005
Matrix inverse is the triangle of Salie numbers A098435. - Paul Barry, Jan 21 2005
Coefficients of Morgan-Voyce polynomial b(n,x); e.g., b(3,x)=x^3+5x^2+6x+1. See A172431 for coefficients of Morgan-Voyce polynomial B(n,x). - Clark Kimberling, Feb 13 2010
T(n,k) is the number of stack polyominoes of perimeter 2n+4 with k+1 columns. - Emanuele Munarini, Apr 07 2011
Roots of signed n-th polynomials are chaotic with respect to the operation (-2, x^2), with cycle lengths A003558(n). Example: starting with a root to x^3 - 5x^2 + 6x - 1 = 0; (2 + 2*cos(2*Pi/N) = 3.24697... = A116415; we obtain the trajectory (3.24697...-> 1.55495...-> 0.198062...; the 3 roots to the polynomial with cycle length 3 matching A003558(3) = 3. The operation (-2, x^2) is the reversal of the well known chaotic operation (x^2 - 2) [Kappraff, Adamson, 2004] starting with seed 2*cos(2*Pi/N). Check: given 2*cos(2*Pi/7) = 1.24697..., we obtain the 3-cycle using (x^2 - 2): (1.24697...-> -0.445041...-> 1.801937...; where the terms in either set are intermediate terms in the other, irrespective of sign. - Gary W. Adamson, Sep 22 2011
A054142 is jointly generated with A172431 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+v(n-1,x) and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section of A172431. - Clark Kimberling, Mar 09 2012
Subtriangle of the triangle given by (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 2012
The o.g.f. for row n of the array A(n, k) = binomial(2*n-k,k), k >= 0, n >= 0 is G(n,x) = Sum_{k=0..n} T(n, k)*x^k + (-x)^(2*n+1) * c(-x)^(2*n+1) / sqrt(1-4*(-x)), for n >= 0. Here c(x) is the o.g.f. of A000108 (Catalan). For powers of c(x) see the W. Lang link in A115139. For the alternating sign case replace x by -x. - Wolfdieter Lang, Sep 12 2016
Multiplying the n-th diagonal by A001147(n) generates A001497. - Tom Copeland, Oct 04 2016

			Triangle begins:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  6,   1;
  1,  7, 15,  10,   1;
  1,  9, 28,  35,  15,   1;
  1, 11, 45,  84,  70,  21,   1;
  1, 13, 66, 165, 210, 126,  28,  1;
  1, 15, 91, 286, 495, 462, 210, 36, 1; ...
...
(0, 1, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 3,  1;
  0, 1, 5,  6,  1;
  0, 1, 7, 15, 10,  1;
  0, 1, 9, 28, 35, 15, 1. _Philippe Deléham_, Apr 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..495
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
Molly Fenn and Eric Sommers, A transitivity result for ad-nilpotent ideals in type A, arXiv:2101.04091 [math.RT], 2021.
J. L. Jacobsen, and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760, arXiv:cond-mat/0407444 See Eq. 2.27. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, 5th Interdispl Symm. Congress and Exh. Jul 8-14, Sydney, 2001 - [with commercial pop-ups].
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Journal of Dynamical Systems and Geometric Theories, Vol. 2 pp. 79-94, (Nov 2004).

These are the even-indexed rows of A011973, the odd-indexed rows form A053123.
Cf. A000108, A003558, A027989, A052535, A054142, A076756, A084938, A085478, A098435, A115139, A172431, A172991, A188648.
Cf. A001147, A001497.

Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n-k,k) ))); # G. C. Greubel, Aug 01 2019

[Binomial(2*n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

T:=(n,k)->binomial(2*n-k,k): seq(seq(T(n,k), k=0..n), n=0..11);

Flatten[Table[Binomial[2n - k, k], {n, 0, 11}, {k, 0, n}]] (* Emanuele Munarini, Apr 07 2011 *)

create_list(binomial(2*n-k,k),n,0,10,k,0,n); /* Emanuele Munarini, Apr 07 2011 */

T(n,k)=if(n<0,0,polcoeff(charpoly(matrix(n,n,i,j,-min(i,j))),k))

[[binomial(2*n-k,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

G.f.: (1-t*z)/((1-t*z)^2-z). - Emeric Deutsch, May 31 2004
Column k has g.f.: (Sum_{j=0..k+1} binomial(k+1, 2j)*x^j)*x^k/(1-x)^(k+1). - Paul Barry, Jun 22 2005
Recurrence: T(n+2,k+2) = T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k). - Emanuele Munarini, Apr 07 2011
T(n, k) = binomial(2*n-k, k) = A085478(n, n-k), for n >= 0, k = 0..n. - Wolfdieter Lang, Mar 25 2020

A188648 Binomial sums a(n) = Sum_{k=0..n} (binomial(2n-k,k))^2.

Original entry on oeis.org

1, 2, 11, 63, 376, 2317, 14545, 92512, 594169, 3844787, 25027296, 163701327, 1075049011, 7083830648, 46812088751, 310118453573, 2058919125662, 13695571200353, 91254952276859, 608960974528058, 4069232436916151

Offset: 0

Emanuele Munarini, Apr 07 2011

Central coefficients of A172991.

Bisection of A051286 (Whitney number of level n of the lattice of the ideals of the fence of order 2n). - Paul D. Hanna, Apr 07 2011

Seiichi Manyama, Table of n, a(n) for n = 0..1198 (terms 0..121 from Vincenzo Librandi)

Sum_{k=0..n} (binomial(2n-k,k))^b: A122367(n) = A001519(n+1) (b=1), this sequence (b=2).

Cf. A054142, A172991, A051286.

Table[Sum[Binomial[2n-k,k]^2,{k,0,n}],{n,0,20}]
Table[DifferenceRoot[Function[{y, m}, {4 (-m + n)^2 (-1 - 2 m + 2 n)^2 y[m] + (-5 m^2 - 18 m^3 - 17 m^4 + 12 m n + 56 m^2 n + 68 m^3 n - 8 n^2 - 56 m n^2 - 100 m^2 n^2 + 16 n^3 + 64 m n^3 - 16 n^4) y[1 + m] + (1 + m)^2 (-m + 2 n)^2 y[2 + m] == 0, y[0] == 0, y[1] == 1}]][n + 1], {n, 0, 20}] (* Benedict W. J. Irwin, Nov 03 2016 *)

makelist(sum(binomial(2*n-k,k)^2,k,0,n),n,0,20);

{a(n) = sum(k=0, n, binomial(2*n-k, k)^2)} \\ Seiichi Manyama, Jan 13 2019

G.f.: 1/2*(1/sqrt(1-2*sqrt(x)-x-2*x*sqrt(x)+x^2) + 1/sqrt(1+2*sqrt(x)-x+2*x*sqrt(x)+x^2)).
Recurrence: (n-2)*n*(2*n - 1)*(48*n^2 - 192*n + 169)*a(n) = (576*n^5 - 4032*n^4 + 10212*n^3 - 11414*n^2 + 5457*n - 849)*a(n-1) + 5*(2*n - 3)*(48*n^4 - 288*n^3 + 565*n^2 - 399*n + 64)*a(n-2) + (576*n^5 - 4608*n^4 + 13668*n^3 - 18286*n^2 + 10521*n - 1896)*a(n-3) - (n-3)*(n-1)*(2*n - 5)*(48*n^2 - 96*n + 25)*a(n-4). - Vaclav Kotesovec, Mar 02 2014
a(n) ~ phi^(4*n + 2) / (2^(3/2) * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 02 2014, simplified Jan 13 2019
Conjecture: a(n) = hypergeom([-n,-n,n+1,n+1], [1/2,1/2,1], 1/16). - Velin Yanev, Oct 31 2019
a(n) = A051286(2*n). - Mark van Hoeij, Sep 05 2022

A027989 a(n) = self-convolution of row n of array T given by A027926.

Original entry on oeis.org

1, 3, 10, 33, 105, 324, 977, 2895, 8462, 24465, 70101, 199368, 563425, 1583643, 4430290, 12342849, 34262337, 94800780, 261545777, 719697255, 1975722326, 5412138033, 14796520365, 40380240528, 110016825025, 299285288499, 813011578522, 2205652007265, 5976479585817

Offset: 0

Clark Kimberling

a(n) is the number of all columns in stack polyominoes of perimeter 2n+4. - Emanuele Munarini, Apr 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A027926, A054142, A172991, A188648.

Table[((5+4n)Fibonacci[1+2n]-(1+2n)Fibonacci[2n])/5,{n,0,28}] (* Emanuele Munarini, Apr 07 2011 *)

makelist(((5+4*n)*fib(1+2*n)-(1+2*n)*fib(2*n))/5,n,0,20); /* Emanuele Munarini, Apr 07 2011 */

Vec((1-3*x+3*x^2)/(1-3*x+x^2)^2+O(x^66)) /* Joerg Arndt, Apr 08 2011 */

a(n) = (2/5)*(n + 1)*F(2*n+3) + (1/5)*F(2*n+2) - (4/5)*(n + 1)*F(2*n), where F(n) = A000045(n). - Ralf Stephan, May 13 2004
From Emanuele Munarini, Apr 07 2011: (Start)
a(n) = ((4*n + 5)*F(2*n+1) - (2*n + 1)*F(2*n))/5, where F(n) = A000045(n).
a(n) = Sum_{k=0..n} binomial(2*n-k, k)*(k + 1).
G.f.: (1 - 3*x + 3*x^2)/(1 - 3*x + x^2)^2.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). (End)

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A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

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A188648 Binomial sums a(n) = Sum_{k=0..n} (binomial(2n-k,k))^2.

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Mathematica

Maxima

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A027989 a(n) = self-convolution of row n of array T given by A027926.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula