0,6

Comment from Paul D. Hanna (pauldhanna(AT)juno.com), 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. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 03 2009]

Triangle begins:

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 program from R. J. Mathar) 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;

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 right-most diagonal which is A000110, the Bell numbers.

A160185 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 03 2009]

Sequence in context: A068098 A135722 A049513 this_sequence A097084 A143327 A094954

Adjacent sequences: A121204 A121205 A121206 this_sequence A121208 A121209 A121210

nonn,tabl

N. J. A. Sloane (njas(AT)research.att.com), based on email from R. J. Mathar, Dec 11 2006