A132677 -id:A132677 - cp's OEIS frontend

A207606 Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.

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

Offset: 1

Clark Kimberling, Feb 19 2012

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

			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;

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

Cf. A207607.

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

(* 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 *)

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

@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

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

A200067 Maximum sum of all products of absolute differences and distances between element pairs among the integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 12, 20, 30, 45, 63, 84, 112, 144, 180, 225, 275, 330, 396, 468, 546, 637, 735, 840, 960, 1088, 1224, 1377, 1539, 1710, 1900, 2100, 2310, 2541, 2783, 3036, 3312, 3600, 3900, 4225, 4563, 4914, 5292, 5684, 6090, 6525, 6975, 7440, 7936, 8448

Offset: 0

Alois P. Heinz, Nov 13 2011

Also the maximum sum of weighted inversions among the compositions of n where weights are products of absolute differences and distances between the element pairs which are not in sorted order.

a(n) is divisible by at least one triangular number >1 for n>=4. Thus 3 is the only prime in this sequence.

			a(2) =  0: [1,1]-> 0, [2]-> 0; the maximum is 0.
a(3) =  1: [1,1,1]-> 0, [2,1]-> 1, [3]-> 0; the maximum is 1.
a(4) =  3: [1,1,1,1]-> 0, [2,1,1]-> 1+2 = 3, [2,2]->0, [3,1]->2, [4]->0.
a(5) =  6: [2,1,1,1]-> 1+2+3 = 6, [3,1,1]-> 2 + 2*2 = 2*(1+2) = 6.
a(6) = 12: [3,1,1,1]-> 2 + 2*2 + 2*3 = 2*(1+2+3) = 12.
a(7) = 20: [3,1,1,1,1]-> 2 + 2*2 + 2*3 + 2*4 = 2*(1+2+3+4) = 20.
a(8) = 30: [3,1,1,1,1,1]-> 2*(1+2+3+4+5) = 30, [4,1,1,1,1]-> 3*(1+2+3+4) = 30.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A000217, A004396, A200068.

a:= n-> (k-> (n-k-1)*k*(k+1)/2)(max(0, floor((2*n-1)/3))):
seq(a(n), n=0..50);

a[n_] := Max[Table[(n-k-1)*k*(k+1)/2, {k, 0, n}]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 22 2013, after Alois P. Heinz *)

G.f.: x^3*(1+x)*(1+x^2)/((1+x+x^2)^2*(x-1)^4).
a(n) = max_{k=0..n} (n-k-1)*k*(k+1)/2.
a(n) = (n-k-1)*k*(k+1)/2 with k = max(0, floor((2*n-1)/3)), or k = A004396(n-1) for n>0.
27*a(n) = (2*n-1)*(n^2-n-1) - A132677(n) - 3*(-1)^n*A099254(n-1). - R. J. Mathar, Mar 14 2025

A191597 Expansion of x(1+3x)/ ( (1-4x)(1+x+x^2)).

Original entry on oeis.org

0, 1, 6, 21, 85, 342, 1365, 5461, 21846, 87381, 349525, 1398102, 5592405, 22369621, 89478486, 357913941, 1431655765, 5726623062, 22906492245, 91625968981, 366503875926, 1466015503701, 5864062014805, 23456248059222, 93824992236885, 375299968947541

Offset: 0

Paul Curtz, Jun 08 2011

a(n) and successive differences define a square array T(0,k) = a(k), T(n,k) = T(n-1,k+1) - T(n-1,k):
0,  1,  6,  21,  85,  342,...
1,  5, 15,  64, 257, 1023,...
4, 10, 49, 193, 766, 3073,...
As with any sequence which obeys a homogeneous linear recurrence (we say it once, only once and we shall not repeat it), the recurrence is also valid for the rows of such arrays of higher order differences.

Index entries for linear recurrences with constant coefficients, signature (3,3,4).

A061347 := proc(n) op(1+(n mod 3),[-2,1,1]) ; end proc:
A191597 := proc(n) (4^n-A061347(n+1))/3 ; end proc:
seq(A191597(n),n=0..30) ; # R. J. Mathar, Jun 08 2011

a(n)=([0,1,0; 0,0,1; 4,3,3]^n*[0;1;6])[1,1] \\ Charles R Greathouse IV, Jul 06 2017

a(n) = 3*a(n-1) + 3*a(n-2) + 4*a(n-3), n >= 3.
a(n) = A024495(2*n).
a(n) = A113405(2*n) + A113405(2*n+1).
a(n+1) - 4*a(n) = A132677(n).
a(n+3) - a(n) = 21*4^n.
a(n) = A178872(n) + 3*A178872(n-1) = (4^n-A061347(n+1))/3. - R. J. Mathar, Jun 08 2011

cp's OEIS Frontend

A207606 Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.

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A200067 Maximum sum of all products of absolute differences and distances between element pairs among the integer partitions of n.

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A191597 Expansion of x(1+3x)/ ( (1-4x)(1+x+x^2)).

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cp's OEIS Frontend

A207606 Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.

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A200067 Maximum sum of all products of absolute differences and distances between element pairs among the integer partitions of n.

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Maple

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A191597 Expansion of x*(1+3*x)/ ( (1-4*x)*(1+x+x^2)).

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A191597 Expansion of x(1+3x)/ ( (1-4x)(1+x+x^2)).