cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A001850 Central Delannoy numbers: a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k).

Original entry on oeis.org

1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823, 252055236609, 1425834724419, 8079317057869, 45849429914943, 260543813797441, 1482376214227923, 8443414161166173, 48141245001931263
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of paths from (0,0) to (n,n) in an n X n grid using only steps north, northeast and east (i.e., steps (1,0), (1,1), and (0,1)).
Also the number of ways of aligning two sequences (e.g., of nucleotides or amino acids) of length n, with at most 2*n gaps (-) inserted, so that while unnecessary gappings: - -a a- - are forbidden, both b- and -b are allowed. (If only other of the latter is allowed, then the sequence A000984 gives the number of alignments.) There is an easy bijection from grid walks given by Dickau to such set of alignments (e.g., the straight diagonal corresponds to the perfect alignment with no gaps). - Antti Karttunen, Oct 10 2001
Also main diagonal of array A008288 defined by m(i,1) = m(1,j) = 1, m(i,j) = m(i-1,j-1) + m(i-1,j) + m(i,j-1). - Benoit Cloitre, May 03 2002
So, as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= n from any given point. These terms occur in the crystal ball sequences: a(n) here is the n-th term in the sequence for the n-dimensional cubic lattice. See A008288 for a list of crystal ball sequences (rows or columns of A008288). - Shel Kaphan, Dec 26 2022
a(n) is the number of n-matchings of a comb-like graph with 2*n teeth. Example: a(2) = 13 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has 13 2-matchings: any of the six possible pairs of teeth and {Aa, BC}, {Aa, CD}, {Bb, CD}, {Cc, AB}, {Dd, AB}, {Dd, BC}, {AB, CD}. - Emeric Deutsch, Jul 02 2002
Number of ordered trees with 2*n+1 edges, having root of odd degree, nonroot nodes of outdegree at most 2 and branches of odd length. - Emeric Deutsch, Aug 02 2002
The sum of the first n coefficients of ((1 - x) / (1 - 2*x))^n is a(n-1). - Michael Somos, Sep 28 2003
Row sums of A063007 and A105870. - Paul Barry, Apr 23 2005
The Hankel transform (see A001906 for definition) of this sequence is A036442: 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005
Also number of paths from (0,0) to (n,0) using only steps U = (1,1), H = (1,0) and D =(1,-1), U can have 2 colors and H can have 3 colors. - N-E. Fahssi, Jan 27 2008
Equals row sums of triangle A152250 and INVERT transform of A109980: (1, 2, 8, 36, 172, 852, ...). - Gary W. Adamson, Nov 30 2008
Number of overpartitions in the n X n box (treat a walk of the type in the first comment as an overpartition, by interpreting a NE step as N, E with the part thus created being overlined). - William J. Keith, May 19 2017
Diagonal of rational functions 1/(1 - x - y - x*y), 1/(1 - x - y*z - x*y*z). - Gheorghe Coserea, Jul 03 2018
Dimensions of endomorphism algebras End(R^{(n)}) in the Delannoy category attached to the oligomorphic group of order preserving self-bijections of the real line. - Noah Snyder, Mar 22 2023
a(n) is the number of ways to tile a strip of length n with white squares, black squares, and red dominos, where we must have an equal number of white and black squares. - Greg Dresden and Leo Zhang, Jul 11 2025

Examples

			G.f. = 1 + 3*x + 13*x^2 + 63*x^3 + 321*x^4 + 1683*x^5 + 8989*x^6 + ...

References

  • Frits Beukers, Arithmetic properties of Picard-Fuchs equations, Séminaire de Théorie des nombres de Paris, 1982-83, Birkhäuser Boston, Inc.
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 593.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
  • L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math., 26 (1961), 223-229.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 2, 1999; see Example 6.3.8 and Problem 6.49.
  • D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 28.

Links

Crossrefs

Cf. A008288, bisection of A026003, A027618, A047665, A052141, A084773, A152250, A109980, A000129, A078057, A241023, A243949.
Main diagonal of A064861.
Column k=2 of A262809 and A263159.

Programs

  • Maple
    seq(add(multinomial(n+k,n-k,k,k),k=0..n),n=0..20); # Zerinvary Lajos, Oct 18 2006
seq(orthopoly[P](n,3), n=0..100); # Robert Israel, Nov 03 2015
  • Mathematica
    f[n_] := Sum[ Binomial[n, k] Binomial[n + k, k], {k, 0, n}]; Array[f, 21, 0] (* Or *)
a[0] = 1; a[1] = 3; a[n_] := a[n] = (3(2 n - 1)a[n - 1] - (n - 1)a[n - 2])/n; Array[a, 21, 0] (* Or *)
CoefficientList[ Series[1/Sqrt[1 - 6x + x^2], {x, 0, 20}], x] (* Robert G. Wilson v *)
Table[LegendreP[n, 3], {n, 0, 22}] (* Jean-François Alcover, Jul 16 2012, from first formula *)
a[n_] := Hypergeometric2F1[-n, n+1, 1, -1]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 26 2013 *)
a[ n_] := With[ {m = If[n < 0, -1 - n, n]}, SeriesCoefficient[ (1 - 6 x + x^2)^(-1/2), {x, 0, m}]]; (* Michael Somos, Jun 10 2015 *)
  • Maxima
    a(n):=coeff(expand((1+3*x+2*x^2)^n),x,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Mar 02 2011 */
  • PARI
    {a(n) = if( n<0, n = -1 - n); polcoeff( 1 / sqrt(1 - 6*x + x^2 + x * O(x^n)), n)}; /* Michael Somos, Sep 23 2006 */
  • PARI
    {a(n) = if( n<0, n = -1 - n); subst( pollegendre(n), x, 3)}; /* Michael Somos, Sep 23 2006 */
  • PARI
    {a(n) = if( n<0, n = -1 - n); n++; subst( Pol(((1 - x) / (1 - 2*x) + O(x^n))^n), x, 1);} /* Michael Somos, Sep 23 2006 */
  • PARI
    a(n)=if(n<0, 0, polcoeff((1+3*x+2*x^2)^n, n)) \\ Paul Barry, Aug 22 2007
  • PARI
    /* same as in A092566 but use */
steps=[[1,0], [0,1], [1,1]]; /* Joerg Arndt, Jun 30 2011 */
  • PARI
    a(n)=sum(k=0,n,binomial(n,k)*binomial(n+k,k)); \\ Joerg Arndt, May 11 2013
  • PARI
    my(x='x+O('x^30)); Vec(1/sqrt(1 - 6*x + x^2)) \\ Altug Alkan, Oct 17 2015
  • Python
    # from Nick Hobson.
def f(a, b):
    if a == 0 or b == 0:
        return 1
    return f(a, b - 1) + f(a - 1, b) + f(a - 1, b - 1)
[f(n, n) for n in range(7)]
  • Python
    from gmpy2 import divexact
A001850 = [1, 3]
for n in range(2,10**3):
    A001850.append(divexact(A001850[-1]*(6*n-3)-(n-1)*A001850[-2],n))
# Chai Wah Wu, Sep 01 2014
  • Sage
    a = lambda n: hypergeometric([-n, -n], [1], 2)
[simplify(a(n)) for n in range(23)] # Peter Luschny, Nov 19 2014

Formula

a(n) = P_n(3), where P_n is n-th Legendre polynomial.
G.f.: 1 / sqrt(1 - 6*x + x^2).
a(n) = a(n-1) + 2*A002002(n) = Sum_{j} A063007(n, j). - Henry Bottomley, Jul 02 2001
Dominant term in asymptotic expansion is binomial(2*n, n)/2^(1/4)*((sqrt(2) + 1)/2)^(2*n + 1)*(1 + c_1/n + c_2/n^2 + ...). - Michael David Hirschhorn
a(n) = Sum_{i=0..n} (A000079(i)*A008459(n, i)) = Sum_{i=0..n} (2^i * C(n, i)^2). - Antti Karttunen, Oct 10 2001
a(n) = Sum_{k=0..n} C(n+k, n-k)*C(2*k, k). - Benoit Cloitre, Feb 13 2003
a(n) = Sum_{k=0..n} C(n, k)^2 * 2^k. - Michael Somos, Oct 08 2003
a(n - 1) = coefficient of x^n in A120588(x)^n if n>=0. - Michael Somos, Apr 11 2012
G.f. of a(n-1) = 1 / (1 - x / (1 - 2*x / (1 - 2*x / (1 - x / (1 - 2*x / (1 - x / ...)))))). - Michael Somos, May 11 2012
INVERT transform is A109980. BINOMIAL transform is A080609. BINOMIAL transform of A006139. PSUM transform is A089165. PSUMSIGN transform is A026933. First backward difference is A110170. - Michael Somos, May 11 2012
E.g.f.: exp(3*x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004
a(n) = Sum_{k=0..n} C(2*n-k, n)*C(n, k). - Paul Barry, Apr 23 2005
a(n) = Sum_{k>=n} binomial(k, n)^2/2^(k+1). - Vladeta Jovovic, Aug 25 2006
a(n) = a(-1 - n) for all n in Z. - Michael Somos, Sep 23 2006
D-finite with recurrence: a(-1) = a(0) = 1; n*a(n) = 3*(2*n-1)*a(n-1) - (n-1)*a(n-2). Eq (4) in T. D. Noe's article in JIS 9 (2006) #06.2.7.
Define general Delannoy numbers by (i,j > 0): d(i,0) = d(0,j) = 1 =: d(0,0) and d(i,j) = d(i-1,j-1) + d(i-2,j-1) + d(i-1,j). Then a(k) = Sum_{j >= 0} d(k,j)^2 + d(k-1,j)^2 = A026933(n)+A026933(n-1). This is a special case of the following formula for general Delannoy numbers: d(k,j) = Sum_{i >= 0, p=0..n} d(p, i) * d(n-p, j-i) + d(p-1, i) * d(n-p-1, j-i-1). - Peter E John, Oct 19 2006
Coefficient of x^n in (1 + 3*x + 2*x^2)^n. - N-E. Fahssi, Jan 11 2008
a(n) = A008288(A046092(n)). - Philippe Deléham, Apr 08 2009
G.f.: 1/(1 - x - 2*x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ... (continued fraction). - Paul Barry, May 28 2009
G.f.: d/dx log(1/(1 - x*A001003(x))). - Vladimir Kruchinin, Apr 19 2011
G.f.: 1/(2*Q(0) + x - 1) where Q(k) = 1 + k*(1-x) - x - x*(k + 1)*(k + 2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n) = Sum_{k=0..n} C(n,k) * C(n+k,k). - Joerg Arndt, May 11 2013
G.f.: G(0), where G(k) = 1 + x*(6 - x)*(4*k + 1)/(4*k + 2 - 2*x*(6-x)*(2*k + 1)*(4*k + 3)/(x*(6 - x)*(4*k + 3) + 4*(k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(6 - x)*(2*k - 1)/(x*(6 - x)*(2*k - 1) + 2*(k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(6 - x)*(2*k + 1)/(x*(6 - x)*(2*k + 1) + 2*(k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013
a(n)^2 = Sum_{k=0..n} 2^k * C(2*k, k)^2 * C(n+k, n-k) = A243949(n). - Paul D. Hanna, Aug 17 2014
a(n) = hypergeom([-n, -n], [1], 2). - Peter Luschny, Nov 19 2014
a(n) = Sum_{k=0..n/2} C(n-k,k) * 3^(n-2*k) * 2^k * C(n,k). - Vladimir Kruchinin, Jun 29 2015
a(n) = A049600(n, n-1).
a(n) = Sum_{0 <= j, k <= n} (-1)^(n+j)*C(n,k)*C(n,j)*C(n+k,k)*C(n+k+j,k+j). Cf. A126086 and A274668. - Peter Bala, Jan 15 2020
a(n) ~ c * (3 + 2*sqrt(2))^n / sqrt(n), where c = 1/sqrt(4*Pi*(3*sqrt(2)-4)) = 0.572681... (Banderier and Schwer, 2005). - Amiram Eldar, Jun 07 2020
a(n+1) = 3*a(n) + 2*Sum_{l=1..n} A006318(l)*a(n-l). [Eq. (1.16) in Qi-Shi-Guo (2016)]
a(n) ~ (1 + sqrt(2))^(2*n+1) / (2^(5/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 09 2023
a(n-1) + a(n) = A241023(n) for n >= 1. - Peter Bala, Sep 18 2024
a(n) = Sum_{k=0..n} C(n+k, 2*k) * C(2*k, k). - Greg Dresden and Leo Zhang, Jul 11 2025

Extensions

New name and reference Sep 15 1995
Formula and more references from Don Knuth, May 15 1996

A036442 a(n) = 2^((n-1)*(n+2)/2).

Original entry on oeis.org

1, 4, 32, 512, 16384, 1048576, 134217728, 34359738368, 17592186044416, 18014398509481984, 36893488147419103232, 151115727451828646838272, 1237940039285380274899124224, 20282409603651670423947251286016, 664613997892457936451903530140172288
Offset: 1

Views

Author

Abdallah Rayhan (rayhan(AT)engr.uvic.ca)

Keywords

Comments

Number of redundant paths for a fault-tolerant ATM switch.
Hankel transform (see A001906 for definition ) of A001850, A006139, A084601; also Hankel transform of the sequence 1, 0, 4, 0, 24, 0, 160, 0, 1120, ... (A059304 with interpolated zeros). - Philippe Deléham, Jul 03 2005
Hankel transform of A109980. Unsigned version of A127945. - Philippe Deléham, Dec 11 2008
a(n) = the multiplicative Wiener index of the wheel graph with n+3 vertices. The multiplicative Wiener index of a connected simple graph G is defined as the product of the distances between all pairs of distinct vertices of G. The wheel graph with n+3 vertices has (n+3)(n+2)/2 pairs of distinct vertices, of which 2(n+2) are adjacent; each of the remaining (n+2)(n-1)/2 pairs are at distance 2; consequently, the multiplicative Wiener index is 2^((n-1)(n+2)/2) = a(n). - Emeric Deutsch, Aug 17 2015

Links

Programs

Formula

a(1) = 1, a(n) = a(n-1) * 2^n. - Vincenzo Librandi, Oct 24 2012

A026770 a(n) = T(2n,n), T given by A026769.

Original entry on oeis.org

1, 2, 7, 28, 120, 538, 2493, 11854, 57558, 284392, 1426038, 7241356, 37173304, 192638992, 1006564439, 5297715628, 28061959428, 149491856978, 800425486692, 4305263668514, 23251846197766, 126044501870378, 685569373724964, 3740339567665558, 20463965229643218, 112250484320225118
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when below the diagonal, (1,1). - Alois P. Heinz, Sep 14 2016

Links

Crossrefs

Cf. A000108, A006318, A026781, A104625, A109980.
Cf. A026769, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026778, A026779.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(x + Sqrt(1-4*x) + Sqrt(1-6*x+x^2)) )); // G. C. Greubel, Nov 01 2019
  • Maple
    seq(coeff(series(2/(x + sqrt(1-4*x) + sqrt(1-6*x+x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 01 2019
  • Mathematica
    T[n_, k_] := T[n, k] = Which[k==0 || k==n, 1, n==2 && k==1, 2, k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], True, T[n-1, k-1] + T[n-1, k]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 24 2019 *)
  • PARI
    { C = (1-sqrt(1-4*x+O(x^51)))/2/x; S = (1-x-sqrt(1-6*x+x^2 +O(x^51)))/2/x; Vec(1/(1-x*(C+S))) } /* Max Alekseyev, Dec 02 2015 */
  • Sage
    def A026770_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2/(x + sqrt(1-4*x) + sqrt(1-6*x+x^2)) ).list()
A026770_list(30) # G. C. Greubel, Nov 01 2019

Formula

O.g.f.: 1/(1-x*(C(x)+S(x))), where C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108 and S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318. - Max Alekseyev, Dec 02 2015

A172094 The Riordan square of the little Schröder numbers A001003.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 17, 7, 1, 45, 76, 40, 10, 1, 197, 353, 216, 72, 13, 1, 903, 1688, 1145, 458, 113, 16, 1, 4279, 8257, 6039, 2745, 829, 163, 19, 1, 20793, 41128, 31864, 15932, 5558, 1356, 222, 22, 1, 103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1
Offset: 0

Views

Author

Philippe Deléham, Jan 25 2010

Keywords

Comments

The Riordan square is defined in A321620.
Previous name was: Triangle, read by rows, given by [1,2,1,2,1,2,1,2,1,2,1,2,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Riordan array (f(x), f(x)-1) where f(x) is the g.f. of A001003. Equals A122538*A007318.

Examples

			Triangle begins:
     1
     1,      1
     3,      4,      1
    11,     17,      7,     1
    45,     76,     40,    10,     1
   197,    353,    216,    72,    13,     1
   903,   1688,   1345,   458,   113,    16,    1
  4279,   8257,   6039,  2745,   829,   163,   19,   1
20793,  41128,  31864, 15932,  5558,  1356,  222,  22,  1
103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1
.
Production matrix begins:
1, 1
2, 3, 1
0, 2, 3, 1
0, 0, 2, 3, 1
0, 0, 0, 2, 3, 1
0, 0, 0, 0, 2, 3, 1
0, 0, 0, 0, 0, 2, 3, 1
0, 0, 0, 0, 0, 0, 2, 3, 1
... - _Philippe Deléham_, Sep 24 2014

Links

Crossrefs

T(n, 0) = A001003(n) (little Schröder), A109980 (row sums).
Diagonals: A239204, A000012, A016777.
Cf. A122538, A007318, A321620.

Programs

  • Maple
    T := (n, k) -> local j; add((binomial(n-1, j)*binomial(n+1, k+j+1) - binomial(n, j)*binomial(n, k+j+1))*2^j, j = 0..n-k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Jan 24 2025
  • Mathematica
    DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x r[[k+1]] + y s[[k+1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k] p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]];
nmax = 9;
DELTA[Table[{1, 2}, (nmax+1)/2] // Flatten, Prepend[Table[0, {nmax}], 1], nmax] // Flatten (* Jean-François Alcover, Aug 07 2018 *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[(1 + x - Sqrt[1 - 6x + x^2])/(4x), 11] // Flatten  (* Peter Luschny, Nov 27 2018 *)

Formula

T(0, 0) = 1, T(n, k) = 0 if k>n, T(n, 0) = T(n-1, 0) + 2*T(n-1, 1), T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k+1) for k>0.
Sum_{0<=k<=n} T(n, k) = A109980(n).
Sum_{k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A001003(m+n).
T(n, k) = Sum_{j=0..n-k} (binomial(n-1, j)*binomial(n+1, k+j+1) - binomial(n, j)*binomial(n, k+j+1))*2^j. (Cigler) - Peter Luschny, Jan 24 2025

Extensions

New name by Peter Luschny, Nov 27 2018

A152250 Eigentriangle, row sums = A001850, the Delannoy numbers.

Original entry on oeis.org

1, 2, 1, 8, 2, 3, 36, 8, 6, 13, 172, 36, 24, 26, 63, 852, 172, 108, 104, 126, 321, 4324, 852, 516, 468, 504, 642, 1683, 22332, 4324, 2556, 2236, 2268, 2568, 3366, 8989, 116876, 22332, 12972, 11076, 10836, 11556, 13464, 17978, 48639
Offset: 0

Views

Author

Gary W. Adamson, Nov 30 2008

Keywords

Comments

Row sums = A001850, the Delannoy numbers: (1, 3, 13, 63, 321,...).
Sum of n-th row terms = rightmost term of next row.

Examples

			First few rows of the triangle =
  1;
  2, 1;
  8, 2, 3;
  36, 8, 6, 13;
  172, 36, 24, 26, 63;
  852, 172, 108, 104, 126, 321;
  4324, 852, 516, 468, 504, 642, 1683;
  22332, 4324, 2556, 2236, 2268, 2568, 3366, 8989;
  116876, 22332, 12972, 11076, 10836, 11556, 13464, 17978, 48639;
  ...
Row 3 = (36, 8, 6, 13) = termwise products of (36, 8, 2, 1) and (1, 1, 3, 13).

Links

Crossrefs

Cf. A001850, A109980.

Programs

  • Mathematica
    nmax = 8;
T[0, 0] = 1;
T[n_, 0] := SeriesCoefficient[1/(x + Sqrt[1 - 6x + x^2]), {x, 0, n}];
T[n_, n_] :=  LegendreP[n - 1, 3];
row[n_] := row[n] = Table[T[m, 0], {m, n, 0, -1}]*Table[T[m, m], {m, 0, n} ];
T[n_, k_] /; 0 < k < n := row[n][[k + 1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

Formula

Triangle read by rows, M*Q. M = an infinite lower triangular matrix with A109980 in every column: (1, 2, 8, 36, 172,...); Q = a matrix with A001850 prefaced with a "1" as the main diagonal: (1, 1, 3, 13, 63, 321,...) and the rest zeros.

A109979 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n, having k (1,1)-steps on the line y=x (a Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps (E=1,0), N=(0,1) and D(1,1)).

Original entry on oeis.org

1, 2, 1, 8, 4, 1, 36, 20, 6, 1, 172, 104, 36, 8, 1, 852, 552, 212, 56, 10, 1, 4324, 2968, 1236, 368, 80, 12, 1, 22332, 16104, 7164, 2336, 580, 108, 14, 1, 116876, 87976, 41372, 14512, 3980, 856, 140, 16, 1, 618084, 483192, 238356, 88848, 26372, 6312, 1204, 176
Offset: 0

Views

Author

Emeric Deutsch, Jul 06 2005

Keywords

Comments

Row sums are the central Delannoy numbers (A001850).
First column yields A109980.
sum(k*T(n,k),k=0..n) = A001109(n).

Examples

			T(2,1)=4 because we have DNE, DEN, NED and END.
Triangle begins:
1;
2,1;
8,4,1;
36,20,6,1;

Links

Crossrefs

Cf. A001850, A109980, A001109.

Programs

  • Maple
    G:=(t*z-z+sqrt(1-6*z+z^2))/(1-6*z+2*t*z^2-t^2*z^2): 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

Formula

G.f.: [tz-z+sqrt(1-6z+z^2)]/(1-6z+2tz^2-t^2*z^2).
Showing 1-6 of 6 results.

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