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A121207 Triangle read by rows. The definition is by diagonals. The r-th diagonal from the right, for r >= 0, is given by b(0) = b(1) = 1; b(n+1) = Sum_{k=0..n} binomial(n+2,k+r)*a(k). 11
1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 9, 15, 1, 1, 5, 14, 31, 52, 1, 1, 6, 20, 54, 121, 203, 1, 1, 7, 27, 85, 233, 523, 877, 1, 1, 8, 35, 125, 400, 1101, 2469, 4140, 1, 1, 9, 44, 175, 635, 2046, 5625, 12611, 21147, 1, 1, 10, 54, 236, 952, 3488, 11226, 30846, 69161, 115975 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

From Paul D. Hanna, Dec 12 2006: (Start)

Consider the row reversal, which is A124496 with an additional left column (A000110 = Bell numbers). The matrix inverse of this triangle is very simple:

1;

-1, 1;

-1, -1, 1;

-1, -2, -1, 1;

-1, -3, -3, -1, 1;

-1, -4, -6, -4, -1, 1;

-1, -5, -10, -10, -5, -1, 1;

-1, -6, -15, -20, -15, -6, -1, 1;

-1, -7, -21, -35, -35, -21, -7, -1, 1;

-1, -8, -28, -56, -70, -56, -28, -8, -1, 1; ...

This gives the recurrence and explains why the Bell numbers appear. (End)

Triangle A160185 = reversal then deletes right border of 1's. - Gary W. Adamson, May 03 2009

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

H. W. Gould and Jocelyn Quaintance, A linear binomial recurrence and the Bell numbers and polynomials. Applicable Analysis and Discrete Mathematics, 1 (2007), 371-385.

EXAMPLE

Triangle begins (compare also table 9.2 in the Gould-Quaintance reference):

1,

1, 1,

1, 1, 2,

1, 1, 3, 5,

1, 1, 4, 9, 15,

1, 1, 5, 14, 31, 52,

1, 1, 6, 20, 54, 121, 203,

1, 1, 7, 27, 85, 233, 523, 877,

1, 1, 8, 35, 125, 400,1101,2469,4140,

1, 1, 9, 44, 175, 635,2046,5625,12611,21147,

1, 1, 10, 54, 236, 952,3488,11226,30846,69161,115975,

1, 1, 11, 65, 309,1366,5579,20425,65676,180474,404663,678570,

1, 1, 12, 77, 395,1893,8494,34685,126817,407787,1120666,2512769,4213597,

1, 1, 13, 90, 495,2550,12432,55818,227550,831915,2675410,7352471,16485691,27644437, etc.

MAPLE

Gould := proc(n, d) local k; if n<=1 then RETURN(1); else

# This is the Jovovic formula with general index 'd'

# where A040027, A045499, etc. use one explicit integer

# Index n+1 is shifted to n from the original formula.

RETURN(add(binomial(n-1+d, k+d)*Gould(k, d), k=0..n-1));

fi;

end:

# row and col refer to the extrapolated super-table:

for row from 0 to 13 do

# working up to row, not row-1, shows also the Bell numbers

# at the end of each row

for col from 0 to row do

# 'diag' is constant for one of A040027, A045499 etc

diag := row-col;

printf("%4d, ", Gould(col, diag));

od;

print();

od;

# R. J. Mathar

# second Maple program:

T:= proc(n, k) option remember; `if`(k=0, 1,

      add(T(n-j, k-j)*binomial(n-1, j-1), j=1..k))

    end:

seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jan 08 2018

MATHEMATICA

g[n_ /; n <= 1, _] := 1; g[n_, d_] := g[n, d] = Sum[ Binomial[n-1+d, k+d]*g[k, d], {k, 0, n-1}]; Flatten[ Table[ diag = row-col; g[col, diag], {row, 0, 13}, {col, 0, row}]] (* Jean-Fran├žois Alcover, Nov 25 2011, after R. J. Mathar *)

PROG

(Python)

# Computes the n-th diagonal of the triangle reading from the right.

from itertools import accumulate

def Gould_diag(diag, size):

    if size < 1: return []

    if size == 1: return [1]

    L, accu = [1, 1], [1]*diag

    for _ in range(size-2):

        accu = list(accumulate([accu[-1]] + accu))

        L.append(accu[-1])

return L # Peter Luschny, Apr 24 2016

CROSSREFS

Diagonals, reading from the right, are A000110, A040027, A045501, A045499, A045500.

A124496 is a very similar triangle, obtained by reversing the rows and appending a rightmost diagonal which is A000110, the Bell numbers. See also A046936, A298804.

Cf. A160185. - Gary W. Adamson, May 03 2009

T(2n,n) gives A297924.

Sequence in context: A135722 A258306 A049513 * A097084 A143327 A094954

Adjacent sequences:  A121204 A121205 A121206 * A121208 A121209 A121210

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, based on email from R. J. Mathar, Dec 11 2006

STATUS

approved

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Last modified February 25 13:30 EST 2018. Contains 299654 sequences. (Running on oeis4.)