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A207606 Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section. 5
1, 2, 3, 2, 4, 7, 2, 5, 16, 11, 2, 6, 30, 36, 15, 2, 7, 50, 91, 64, 19, 2, 8, 77, 196, 204, 100, 23, 2, 9, 112, 378, 540, 385, 144, 27, 2, 10, 156, 672, 1254, 1210, 650, 196, 31, 2, 11, 210, 1122, 2640, 3289, 2366, 1015, 256, 35, 2, 12, 275, 1782, 5148, 8008 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

As triangle T(n,k) with 0 <= k <= n, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

LINKS

G. C. Greubel, Rows n = 1..101 of the triangle, flattened

FORMULA

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

As triangle T(n,k) with 0 <= k <= n: g.f.: (1-y*x)/(1-(2+y)*x+x^2). - Philippe Deléham, Mar 03 2012

As triangle T(n,k) with 0 <= k <= n: Sum_{k=0..n} T(n,k)*x^k = A132677(n), A000034(n)*A057077(n), A057079(n), A000027(n+1), A001519(n+1), A001075(n), A002310(n), A038725(n), A172968(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Mar 03 2012

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Mar 03 2012

T(n,k) = C(n+k-1,2*k+1) + 2*C(n+k-1,2*k), where C is binomial. - Yuchun Ji, May 23 2019

T(n,k) = T(n-1,k) + A207607(n-1,k). - Yuchun Ji, May 28 2019

EXAMPLE

First five rows:

  1;

  2;

  3,  2;

  4,  7,  2;

  5, 16, 11,  2;

Triangle (2, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, ...), 0 <= k <= n, begins:

  1;

  2,   0;

  3,   2,   0;

  4,   7,   2,   0;

  5,  16,  11,   2,   0;

  6,  30,  36,  15,   2,   0;

  7,  50,  91,  64,  19,   2,   0;

  8,  77, 196, 204, 100,  23,   2,   0;

MAPLE

T:= proc(n, k) option remember;

      if k<0 or k>n then 0

    elif k=0 then n+2

    elif k=n then 2

    else 2*T(n-1, k) + T(n-1, k-1) - T(n-2, k)

      fi; end:

1, seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Mar 15 2020

MATHEMATICA

(* First program *)

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

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

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

Table[Factor[u[n, x]], {n, 1, z}]

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

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

TableForm[cu]

Flatten[%]  (* A207606 *)

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

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

TableForm[cv]

Flatten[%]  (* A207607 *)

(* Second program *)

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, n+2, If[k==n, 2, 2*T[n-1, k] - T[n-2, k] + T[n-1, k-1] ]]]; Join[{1}, Table[T[n, k], {n, 0, 10}, {k, 0, n}]]//Flatten (* G. C. Greubel, Mar 15 2020 *)

PROG

(Python)

from sympy import Poly

from sympy.abc import x

def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)

def v(n, x): return 1 if n==1 else x*u(n - 1, x) + (x + 1)*v(n - 1, x)

def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]

for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

(Sage)

@CachedFunction

def T(n, k):

    if (k<0 or k>n): return 0

    elif (k==1): return n+1

    elif (k==n): return 2

    else: return 2*T(n-1, k) + T(n-1, k-1) - T(n-2, k)

[1]+[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 15 2020

CROSSREFS

Cf. A207607.

Sequence in context: A341098 A254967 A229012 * A303845 A132439 A338902

Adjacent sequences:  A207603 A207604 A207605 * A207607 A207608 A207609

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Feb 19 2012

STATUS

approved

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Last modified June 24 07:56 EDT 2021. Contains 345416 sequences. (Running on oeis4.)