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A209577 Triangle of coefficients of polynomials u(n,x) jointly generated with A209578; see the Formula section. 3
1, 1, 1, 3, 2, 1, 5, 6, 3, 1, 9, 13, 10, 4, 1, 15, 28, 26, 15, 5, 1, 25, 56, 64, 45, 21, 6, 1, 41, 109, 146, 124, 71, 28, 7, 1, 67, 206, 319, 315, 216, 105, 36, 8, 1, 109, 382, 671, 758, 602, 349, 148, 45, 9, 1, 177, 697, 1372, 1744, 1576, 1056, 533, 201, 55, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

For a discussion and guide to related arrays, see A208510.

LINKS

Table of n, a(n) for n=1..65.

FORMULA

u(n,x) = x*u(n-1,x) + v(n-1,x),

v(n,x) = (x+1)*u(n-1,x) + v(n-1,x) + 1,

where u(1,x)=1, v(1,x)=1.

The coefficients in the triangle seem to be T(n,m) = sum(k=0,n-m,2 * binomial(m+k, m)*binomial(k, n-k-m) - sum(i=0, n-m-k, binomial(m+k-1,k)*binomial(k,n-m-i-k))) by using the PARI syntax. - Thomas Baruchel, Jun 03 2018

EXAMPLE

First five rows:

  1;

  1,  1;

  3,  2,  1;

  5,  6,  3,  1;

  9, 13, 10,  4,  1;

First three polynomials v(n,x): 1, 1 + x, 3 + 2x + x^2.

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];

v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;

Table[Expand[u[n, x]], {n, 1, z/2}]

Table[Expand[v[n, x]], {n, 1, z/2}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]   (* A209577 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]   (* A209578 *)

CROSSREFS

Cf. A209578, A208510.

Sequence in context: A132970 A192022 A208608 * A139377 A138483 A110712

Adjacent sequences:  A209574 A209575 A209576 * A209578 A209579 A209580

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 11 2012

STATUS

approved

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Last modified October 14 07:19 EDT 2019. Contains 327995 sequences. (Running on oeis4.)