A131438 -id:A131438

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A131439

Inverse binomial transform of A131438 (assuming zero offset in both sequences)

+20
3

1, 7, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: The sequence appears to be (1, 7, ...) followed by 4k + 14; k=0,1,2,...; thus: (1, 7, 14, 18, 22, 26, ...).

Inverse binomial transform of this sequence = (1, 6, 1, -4, 7, -10, 13, -16, 19, -22, ...).

LINKS

Table of n, a(n) for n=1..45.

FORMULA

A007318^(-1) * A131438.

a(n) = 2*a(n-1) - a(n-2) for n>4. G.f.: -x*(x+1)*(3*x^2-4*x-1) / (x-1)^2. [Colin Barker, Jan 06 2013]

EXAMPLE

(1, 3, 3, 1) dot (1, 7, 14, 18) = 82 = A131438(4).

CROSSREFS

Cf. A131436, A131437, A131438.

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Jul 11 2007

STATUS

approved

A131436

Triangle read by rows, (n-1) zeros followed by 2^n - 1.

+10
4

1, 0, 3, 0, 0, 7, 0, 0, 0, 15, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 63, 0, 0, 0, 0, 0, 0, 127, 0, 0, 0, 0, 0, 0, 0, 255, 0, 0, 0, 0, 0, 0, 0, 0, 511, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2047, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4095, 0, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

FORMULA

2*A059268(n)-A059268(n+1). - Paul Curtz, Jul 03 2008

EXAMPLE

First few rows of the triangle are:

0, 3;

0, 0, 7;

0, 0, 0, 15;

0, 0, 0, 0, 31;

...

MATHEMATICA

Flatten[Table[{PadRight[{}, n-1, 0], 2^n-1}, {n, 20}]] (* Harvey P. Dale, May 23 2012 *)

CROSSREFS

Cf. A131437, A131438, A131439.

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jul 11 2007

EXTENSIONS

More terms from Harvey P. Dale, May 23 2012

STATUS

approved

A131437

(A000012 * A131436) + (A131436 * A000012) - A000012.

+10
4

1, 3, 5, 7, 9, 13, 15, 17, 21, 29, 31, 33, 37, 45, 61, 63, 65, 69, 77, 93, 125, 127, 129, 133, 141, 157, 189, 253, 255, 257, 261, 269, 285, 317, 381, 509, 511, 513, 517, 525, 541, 573, 637, 765, 1021, 1023, 1025, 1029, 1037, 1053, 1085, 1149, 1277, 1533, 2045

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Left column = 2^n - 1; right border = A036563, 2^(n+1) - 3: (1, 5, 13, 29, 61, 125, ...). Row sums = A131438: (1, 8, 29, 82, 207, 492, 1129, ...).

LINKS

Table of n, a(n) for n=1..55.

FORMULA

(A000012 * A131436) + (A131436 * A000012) - A000012; as infinite lower triangular matrices.

EXAMPLE

First few rows of the triangle are:

3, 5;

7, 9, 13;

15, 17, 21, 29;

31, 33, 37, 45, 61;

63, 65, 69, 77, 93, 125;

...

MAPLE

A000012 := proc(n, k)

1 ;

end proc:

A131436 := proc(n, k)

if k = n then

2^n-1 ;

else

end if;

end proc:

A131437 := proc(n, k)

add( A000012(n, i)*A131436(i, k) + A131436(n, i)*A000012(i, k), i=k..n) -1 ;

end proc:

seq(seq(A131437(n, k), k=1..n), n=1..15) ; # R. J. Mathar, Sep 24 2011

CROSSREFS

Cf. A036563, A131436, A131438, A131439.

KEYWORD

nonn,easy,tabl

AUTHOR

Gary W. Adamson, Jul 11 2007

EXTENSIONS

Corrected by R. J. Mathar, Sep 24 2011

STATUS

approved

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