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A156050 Triangle T(n,m) = binomial(n,m)+2*P(n,m) read by rows, where P(n,m) = 1+A000041(n)-A000041(m)-A000041(n-m). 3
1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 8, 10, 8, 1, 1, 9, 16, 16, 9, 1, 1, 14, 25, 32, 25, 14, 1, 1, 15, 35, 51, 51, 35, 15, 1, 1, 22, 48, 82, 96, 82, 48, 22, 1, 1, 25, 64, 118, 164, 164, 118, 64, 25, 1, 1, 34, 83, 170, 264, 310, 264, 170, 83, 34, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The element-by-element sum of: the Pascal triangle A007318 plus two times the elements of P(n,m).

Row sums are 2^n+2*( (n+1)*(1+A000041(n)) -2*A000070(n) ), starting 1, 2, 6, 12, 28, 52, 112, 204, 402, 744, 1414,..., always below factorial(n+1).

The remarkable thing about this sub-Eulerian numbers triangle is that it a Sierpinski gasket modulo 2.

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

P(n,m) starts in row n= 0 as

0

0, 0

0, 1, 0

0, 1, 1, 0

0, 2, 2, 2, 0

0, 2, 3, 3, 2, 0

0, 4, 5, 6, 5, 4, 0

0, 4, 7, 8, 8, 7, 4, 0

0, 7, 10, 13, 13, 13, 10, 7, 0

0, 8, 14, 17, 19, 19, 17, 14, 8, 0

0, 12, 19, 25, 27, 29, 27, 25, 19, 12, 0

to yield T(n,m) from row n=0 on:

1,

1, 1,

1, 4, 1,

1, 5, 5, 1,

1, 8, 10, 8, 1,

1, 9, 16, 16, 9, 1,

1, 14, 25, 32, 25, 14, 1,

1, 15, 35, 51, 51, 35, 15, 1,

1, 22, 48, 82, 96, 82, 48, 22, 1,

1, 25, 64, 118, 164, 164, 118, 64, 25, 1,

1, 34, 83, 170, 264, 310, 264, 170, 83, 34, 1

MATHEMATICA

Clear[f];

t[n_, m_] = 1 + PartitionsP[n] - PartitionsP[m] - PartitionsP[n - m]; \! Table[(Table[t[n, m], {m, 0, n}] + Reverse[Table[t[n, m], {m, 0, n}]])/2, {n, 0, 10}];

Table[Table[Binomial[n, m], {m, 0, n}] + (Table[t[n, m], {m, 0, n}] + Reverse[Table[t[n, m], {m, 0, n}]]), {n, 0, 10}];

Flatten[%]

CROSSREFS

Sequence in context: A204621 A146770 A143334 * A136489 A166455 A171142

Adjacent sequences:  A156047 A156048 A156049 * A156051 A156052 A156053

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Feb 02 2009

EXTENSIONS

Row sum formula and P(n,m) examples added - The Assoc. Eds. of the OEIS, Aug 29 2010

STATUS

approved

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Last modified June 18 17:05 EDT 2019. Contains 324214 sequences. (Running on oeis4.)