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A100326 Triangle, read by rows, where row n equals the inverse binomial of column n of square array A100324, which lists the self-convolutions of SHIFT(A003169). 8
1, 1, 1, 3, 4, 1, 14, 20, 7, 1, 79, 116, 46, 10, 1, 494, 736, 311, 81, 13, 1, 3294, 4952, 2174, 626, 125, 16, 1, 22952, 34716, 15634, 4798, 1088, 178, 19, 1, 165127, 250868, 115048, 36896, 9094, 1724, 240, 22, 1, 1217270, 1855520, 862607, 285689, 74687 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The leftmost column equals A003169 shift one place right.

Each column k>0 equals the convolution of the prior column and A003169.

Row sums form A100327.

The elements of the matrix inverse are T^(-1)(n,k) = (-1)^(n+k) * A158687(n,k). - R. J. Mathar, Mar 15 2013

LINKS

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

FORMULA

T(n, 0) = A003169(n) = Sum_{k=0..n-1} (k+1)*T(n-1, k) for n>0, with T(0, 0)=1.

T(n, k) = Sum_{i=0..n-k} T(i+1, 0)*T(n-i-1, k-1) for n>0.

G.f.: A(x, y) = (1 + G(x))/(1 - y*G(x)), where G(x) is the g.f. of A003169.

EXAMPLE

Leftmost column equals Sum_{k=0..n-1} (k+1)*T(n-1,k):

T(4,0) = 79 = 1*(14)+2*(20)+3*(7)+4*(1) = 1*T(3,0)+2*T(3,1)+3*T(3,2)+4*T(3,3).

All other elements are from the convolution of prior column and A003169:

T(4,2) = 46 = 1*(20)+3*(4)+14*(1) = T(1,0)*T(3,1)+T(2,0)*T(2,1)+T(3,0)*T(1,1).

Rows begin:

1;

1, 1;

3, 4, 1;

14, 20, 7, 1;

79, 116, 46, 10, 1;

494, 736, 311, 81, 13, 1;

3294, 4952, 2174, 626, 125, 16, 1;

22952, 34716, 15634, 4798, 1088, 178, 19, 1;

165127, 250868, 115048, 36896, 9094, 1724, 240, 22, 1;

1217270, 1855520, 862607, 285689, 74687, 15629, 2561, 311, 25, 1;

9146746, 13979192, 6567862, 2229322, 608909, 136792, 25051, 3626, 391, 28, 1; ...

First column forms A003169 shift right.

Binomial transform of row 3 forms column 3 of square A100324:

BINOMIAL([14,20,7,1]) = [14,34,61,96,140,194,259,...].

Binomial transform of row 4 forms column 4 of square A100324:

BINOMIAL([79,116,46,10,1]) = [79,195,357,575,860,1224,...].

MAPLE

A100326 := proc(n, k)

    if k < 0 or k > n then

        0 ;

    elif n = 0 then

        1 ;

    elif k = 0 then

        A003169(n)

    else

        add(procname(i+1, 0)*procname(n-i-1, k-1), i=0..n-k) ;

    end if;

end proc: # R. J. Mathar, Mar 15 2013

MATHEMATICA

lim = 9; t[0, 0] = 1; t[n_, 0] := t[n, 0] = Sum[(k + 1)*t[n - 1, k], {k, 0, n - 1}]; t[n_, k_] := t[n, k] = Sum[t[j + 1, 0]*t[n - j - 1, k - 1], {j, 0, n - k}]; Flatten[ Table[ t[n, k], {n, 0, lim}, {k, 0, n}]] (* Jean-Fran├žois Alcover, Sep 20 2011 *)

PROG

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==0, 1, if(k==0, sum(i=0, n-1, (i+1)*T(n-1, i)), sum(i=0, n-k, T(i+1, 0)*T(n-i-1, k-1))); ))

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

(Haskell)

import Data.List (transpose)

a100326 n k = a100326_tabl !! n !! k

a100326_row n = a100326_tabl !! n

a100326_tabl = [1] : f [[1]] where

f xss@(xs:_) = ys : f (ys : xss) where

ys = y : map (sum . zipWith (*) (zs ++ [y])) (map reverse zss)

y = sum $ zipWith (*) [1..] xs

zss@((_:zs):_) = transpose $ reverse xss

-- Reinhard Zumkeller, Nov 21 2015

CROSSREFS

Cf. A003169, A100324, A100327.

Cf. A264717 (central terms).

Sequence in context: A114189 A200659 A059110 * A303728 A321627 A028338

Adjacent sequences:  A100323 A100324 A100325 * A100327 A100328 A100329

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Nov 17 2004

STATUS

approved

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Last modified April 24 19:49 EDT 2019. Contains 322446 sequences. (Running on oeis4.)