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A054336 A convolution triangle of numbers based on A001405 (central binomial coefficients). 13
1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 6, 10, 9, 4, 1, 10, 22, 22, 14, 5, 1, 20, 44, 54, 40, 20, 6, 1, 35, 93, 123, 109, 65, 27, 7, 1, 70, 186, 281, 276, 195, 98, 35, 8, 1, 126, 386, 618, 682, 541, 321, 140, 44, 9, 1, 252, 772, 1362, 1624, 1440, 966, 497, 192, 54, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

T(n,k) is the number of 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) with no level steps at positive height and having k blue level steps. Example: T(4,2)=9 because, denoting U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0), we have BBRR, BRBR, BRRB, RBBR, RBRB, RRBB, BBUD, BUDB, and UDBB. - Emeric Deutsch, Jun 07 2011

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.

The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/(1-(1+x)*z-z^2*c(z^2)), with c(x) the g.f. for Catalan numbers A000108.

Column sequences: A001405, A045621.

Riordan array (f(x), x*f(x)), f(x) the g.f. of A001405. - Philippe Deléham, Dec 08 2009

From Paul Barry, Oct 21 2010: (Start)

Riordan array ((sqrt(1+2x) - sqrt(1-2x))/(2x*sqrt(1-2x)), (sqrt(1+2x)-sqrt(1-2x))/(2sqrt(1-2x))),

inverse of Riordan array ((1+x)/(1+2x+2x^2), x(1+x)/(1+2x+2x^2)) (A181472). (End)

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

G.f. for column m: cbi(x)*(x*cbi(x))^m, with cbi(x) := (1+x*c(x^2))/sqrt(1-4*x^2) = 1/(1-x-x^2*c(x^2)), where c(x) is the g.f. for Catalan numbers A000108.

T(n,k) = Sum_{j>=0} A053121(n,j)*binomial(j,k). - Philippe Deléham, Mar 30 2007

T(n,k) = T(n-1,k-1) + T(n-1,l) + Sum_{j>=0} T(n-1,k+1+j)*(-1)^j. - Philippe Deléham, Feb 23 2012

EXAMPLE

Fourth row polynomial (n=3): p(3,x)= 3 + 5*x + 3*x^2 + x^3.

From Paul Barry, Oct 21 2010: (Start)

Triangle begins

   1;

   1,  1;

   2,  2,   1;

   3,  5,   3,   1;

   6, 10,   9,   4,  1;

  10, 22,  22,  14,  5,  1;

  20, 44,  54,  40, 20,  6, 1;

  35, 93, 123, 109, 65, 27, 7, 1;

Production matrix is

   1,  1;

   1,  1,  1;

  -1,  1,  1,  1;

   1, -1,  1,  1,  1;

  -1,  1, -1,  1,  1,  1;

   1, -1,  1, -1,  1,  1,  1;

  -1,  1, -1,  1, -1,  1,  1, 1;

   1, -1,  1, -1,  1, -1,  1, 1, 1;

  -1,  1, -1,  1, -1,  1, -1, 1, 1, 1; (End)

MATHEMATICA

c[n_, j_] /; n < j || OddQ[n - j] = 0; c[n_, j_] = (j + 1) Binomial[n + 1, (n - j)/2]/(n + 1); t[n_, k_] := Sum[c[n, j]*Binomial[j, k], {j, 0, n}]; Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]][[;; 66]] (* Jean-François Alcover, Jul 13 2011, after Philippe Deléham *)

PROG

(PARI)

A053121(n, k) = if((n-k+1)%2==0, 0, (k+1)*binomial(n+1, (n-k)\2)/(n+1) );

T(n, k) = sum(j=k, n, A053121(n, j)*binomial(j, k));

for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 21 2019

(MAGMA)

A053121:= func< n, k | ((n-k+1) mod 2) eq 0 select 0 else (k+1)*Binomial(n+1, Floor((n-k)/2))/(n+1) >;

T:= func< n, k | (&+[Binomial(j, k)*A053121(n, j): j in [k..n]]) >;

[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 21 2019

(Sage)

def A053121(n, k):

    if (n-k+1) % 2==0: return 0

    else: return (k+1)*binomial(n+1, ((n-k)//2))/(n+1)

def T(n, k): return sum(binomial(j, k)*A053121(n, j) for j in (k..n))

[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 21 2019

(GAP)

A053121:= function(n, k)

    if ((n-k+1) mod 2)=0 then return 0;

    else return (k+1)*Binomial(n+1, Int((n-k)/2))/(n+1);

    fi;

  end;

T:= function(n, k)

    return Sum([k..n], j-> Binomial(j, k)*A053121(n, j));

  end;

Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Jul 21 2019

CROSSREFS

Cf. A001405, A035324, A054335.

Row sums: A054341.

Sequence in context: A139375 A106198 A202847 * A284644 A079956 A140717

Adjacent sequences:  A054333 A054334 A054335 * A054337 A054338 A054339

KEYWORD

easy,nice,nonn,tabl

AUTHOR

Wolfdieter Lang, Mar 13 2000

STATUS

approved

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Last modified April 3 04:21 EDT 2020. Contains 333195 sequences. (Running on oeis4.)