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A155865 A sequence of polynomial coefficients related to the first derivative of the Pascal triangle: p(x,n)=x^n+1+x*d(x+1)^(n+1)/dx=If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n - 1), {x, 1}]]. 0
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 4, 12, 12, 4, 1, 1, 5, 20, 30, 20, 5, 1, 1, 6, 30, 60, 60, 30, 6, 1, 1, 7, 42, 105, 140, 105, 42, 7, 1, 1, 8, 56, 168, 280, 280, 168, 56, 8, 1, 1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Row sums are 1, 2, 3, 6, 14, 34, 82, 194, 450, 1026, 2306,..., essentially 2+A001787(n-1).

LINKS

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

FORMULA

p(x,n)=x^n+1+x*d(x+1)^(n)/dx

p(x,n)=If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n - 1), {x, 1}]]

t(n,m)=coefficients(p(x,n))

Contribution from Roger L. Bagula, Mar 09 2010: (Start)

c(n)=If[n == 0, 1, If[n == 1, 1, Product[(i - 1), {i, 2, n}]]];

t(n,m)=c(n)/(c(m)*c(n-m)) (End)

The triangle is the ConvOffsStoT transform of the natural numbers prefaced with a 1. A row with n integers is the ConvOffs transform of a finite series of the first (n-1) terms in (1, 1, 2, 3, 4,...).  See A214281 for definitions of the transform. - Gary W. Adamson, Jul 09 2012

EXAMPLE

{1},

{1, 1},

{1, 1, 1},

{1, 2, 2, 1},

{1, 3, 6, 3, 1},

{1, 4, 12, 12, 4, 1},

{1, 5, 20, 30, 20, 5, 1},

{1, 6, 30, 60, 60, 30, 6, 1},

{1, 7, 42, 105, 140, 105, 42, 7, 1},

{1, 8, 56, 168, 280, 280, 168, 56, 8, 1},

{1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 1}

...

ConvOffs transform of (1, 1, 2, 3) = integers of row 4: (1, 3, 6, 3, 1). Gary W. Adamson, Jul 09 2012

MATHEMATICA

Clear[p, n, m, x, a];

p[x_, n_] = If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n - 1), {x, 1}]];

Table[ExpandAll[p[x, n]], {n, 0, 10}];

a = Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];

Flatten[a]

Contribution from Roger L. Bagula, Mar 09 2010: (Start)

(* at q=1*)

c[n_, q_] = If[n == 0, 1, If[n == 1, 1, Product[(i - 1)^q, {i, 2, n}]]];

t[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);

Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}] (End)

CROSSREFS

Sequence in context: A008302 A131791 A010358 * A156133 A010048 A055870

Adjacent sequences:  A155862 A155863 A155864 * A155866 A155867 A155868

KEYWORD

nonn,tabl,easy,uned

AUTHOR

Roger L. Bagula, Jan 29 2009

STATUS

approved

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Last modified August 16 09:54 EDT 2018. Contains 313791 sequences. (Running on oeis4.)