A168641 -id:A168641 - cp's OEIS frontend

A168643 Triangle read by rows: T(n,k) = [x^k] p(x,n) where p(x,0) = 1, p(x,n) = (6 - n)(1+x)^n - (5-n)(1 + x^n) for 1 <= n <= 4, and p (x,n) = 4(1+x)^n - Sum_{i=0..2} (Sum_{j=0..i} binomial(n, j)(x^j + x^(n-j))) for n >= 5.

1, 1, 1, 1, 8, 1, 1, 9, 9, 1, 1, 8, 12, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 80, 45, 12, 1, 1, 14, 63, 140, 140, 63, 14, 1, 1, 16, 84, 224, 280, 224, 84, 16, 1, 1, 18, 108, 336, 504, 504, 336, 108, 18, 1, 1, 20, 135, 480, 840, 1008, 840, 480, 135, 20, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

			Triangle begins:
  1;
  1,  1;
  1,  8,   1;
  1,  9,   9,   1;
  1,  8,  12,   8,   1;
  1, 10,  30,  30,  10,    1;
  1, 12,  45,  80,  45,   12,   1;
  1, 14,  63, 140, 140,   63,  14,   1;
  1, 16,  84, 224, 280,  224,  84,  16,   1;
  1, 18, 108, 336, 504,  504, 336, 108,  18,  1;
  1, 20, 135, 480, 840, 1008, 840, 480, 135, 20, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A132046, A168641, A168644, A168646.

(* First program *)
p[x_, n_]:= If[n==0, 1, If[n==1, x+1, 4*(1+x)^n - (1+x^n) - If[n>2, x^n + n*x^(n-1) +n*x+1, 1+x^n] - If[n>3, x^n +n*x^(n-1) + Binomial[n,2]*(x^2 +x^(n-2)) +n*x+1, 1+x^n]]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
(* Second program *)
f[n_,k_]:= With[{B=Boole}, If[n==0, 1, If[0n-3]]]];
A168643[n_,k_]:= Binomial[n,k]*f[n,k];
Table[A168643[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2025 *)

T(n,k) := if k = 0 or k = n then 1 else (if n <= 4 then (6 - n)*binomial(n, k) else ratcoef(4*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 2), x, k))$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */

def f(n,k):
    if n==0: return 1
    elif 0n-3)
def A168643(n,k): return binomial(n,k)*f(n,k)
print(flatten([[A168643(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 08 2025

From G. C. Greubel, Apr 08 2025: (Start)
T(n, k) = [k=0] + (6-n)*binomial(n,k)*[1 <= k <= n-1] + [k=n] if 1 <= n <= 4, T(n, k) = binomial(n,k)*( (k+1)*[k<3] + 4*[2 < k < n-2] + (n-k+1)*[k > n-3] ) if n >= 5, with T(n, 0) = T(n, n) = 1.
T(n, n-k) = T(n, k) (symmetric rows).
Sum_{k=0..n} T(n, k) = 2^(n+2) - n^2 - 3*n - 6 + 13*[n=3] + 10*[n=2] + 4*[n=1] + 3*[n=0]. (End)

Edited by Franck Maminirina Ramaharo, Jan 02 2019

A168644 Triangle read by rows: T(n, k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (7 - n)(1+x)^n - (6-n)(1 + x^n) for 1 <= n <= 5, and p(x,n) = 5(1 + x)^n - Sum_{i=0..3} (Sum_{j=0..i} binomial(n, j)(x^j + x^(n-j))) + (1/6)n(n - 1)(n - 5)x^(n-3) for n >= 6.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 12, 12, 1, 1, 12, 18, 12, 1, 1, 10, 20, 20, 10, 1, 1, 12, 45, 65, 45, 12, 1, 1, 14, 63, 140, 154, 63, 14, 1, 1, 16, 84, 224, 350, 252, 84, 16, 1, 1, 18, 108, 336, 630, 630, 384, 108, 18, 1, 1, 20, 135, 480, 1050, 1260, 1050, 555, 135, 20, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

			Triangle begins:
  1;
  1,  1;
  1, 10,   1;
  1, 12,  12,   1;
  1, 12,  18,  12,    1;
  1, 10,  20,  20,   10,    1;
  1, 12,  45,  65,   45,   12,    1;
  1, 14,  63, 140,  154,   63,   14,   1;
  1, 16,  84, 224,  350,  252,   84,  16,   1;
  1, 18, 108, 336,  630,  630,  384, 108,  18,  1;
  1, 20, 135, 480, 1050, 1260, 1050, 555, 135, 20, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A132046, A168641, A168643, A168646.

(* First program *)
p[x_, n_]:= If[n<2, (1+x)^n, 5*(1+x)^n -(1+x^n) - If[n>2, x^n +n*x^(n-1) + n*x +1, 1+x^n] - If[n>3, x^n +n*x^(n-1) +Binomial[n,2]*(x^2 +x^(n-2)) + n*x +1, 1+x^n] - If[n>4, x^n +n*x^(n-1) +Binomial[n,2]*(x^2 +x^(n-3) +x^(n-2)) + Binomial[n,3]*x^3 +n*x +1, 1+x^n]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
(* Second program *)
f[n_, k_]:= With[{B=Boole}, If[k==0 || k==n, 1, If[1<=n<=5, (7-n) - (6-n)*(B[k==0] + B[k==n]), If[n==6, (k+1)*B[k<4] + (n-k+1)*B[k>3] - B[k==3], (k + 1)*B[k<4] + 5*B[3n-4]]]]];
A168644[n_, k_]:= Binomial[n,k]*f[n,k] + If[n>5, n*(n-1)*(n-5)*Boole[k==n-3]/6, 0];
Table[A168644[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 06 2025 *)

T(n, k) := if k = 0 or k = n then 1 else (if n <= 5 then (7 - n)*binomial(n, k) else ratcoef(5*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 3) + (1/6)*n*(n - 1)*(n - 5)*x^(n - 3), x, k))$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */

def f(n, k):
    if k==0 or k==n: return 1
    elif 03) - int(k==3)
    else: return (k+1)*int(k<4) + 5*int(3n-4)
def A168644(n, k):
    if n<6: return binomial(n, k)*f(n, k)
    else: return binomial(n,k)*f(n,k) + n*(n-1)*(n-5)*int(k==n-3)//6
print(flatten([[A168644(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 06 2025

Edited by Franck Maminirina Ramaharo, Jan 02 2019

A168646 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (8 - n)(1+x)^n - (7 - n)(1 + x^n) for 1 <= n <= 6, and p(x,n) = 6(1+x)^n - Sum_{i=0..4} (Sum_{j=0..i} binomial(n, j)(x^j + x^(n-j))) for n >= 7.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 15, 15, 1, 1, 16, 24, 16, 1, 1, 15, 30, 30, 15, 1, 1, 12, 30, 40, 30, 12, 1, 1, 14, 63, 315, 315, 63, 14, 1, 1, 16, 84, 224, 700, 224, 84, 16, 1, 1, 18, 108, 336, 630, 630, 336, 108, 18, 1, 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1, 1, 22, 165, 660, 1650, 2772, 2772, 1650, 660, 165, 22, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

			Triangle begins:
  1;
  1,  1;
  1, 12,   1;
  1, 15,  15,   1;
  1, 16,  24,  16,    1;
  1, 15,  30,  30,   15,    1;
  1, 12,  30,  40,   30,   12,    1;
  1, 14,  63, 315,  315,   63,   14,   1;
  1, 16,  84, 224,  700,  224,   84,  16,   1;
  1, 18, 108, 336,  630,  630,  336, 108,  18,  1;
  1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A132046, A168641, A168643, A168644.

(* First program *)
p[n_, x_]:= With[{B=Binomial}, If[n==0, 1, If[1<=n<=6, 1 + (8-n)*Sum[B[n,j]*x^j, {j, n -1}] +x^n, Sum[(j+1)*B[n,j]*x^j, {j,0,4}] +6*Sum[B[n,j]*x^j, {j,5,n-5}] + Sum[(n-j+ 1)*B[n,j]*x^j, {j,n-4,n}]]]];
Flatten[Table[CoefficientList[p[n,x], x], {n, 0, 12}]]
(* Second program *)
f[n_, k_]:= If[k==0||k==n,1,If[1<=n<= 6 && 1<=k<=n-1, 8-n, (k+1)*Boole[k<=4] + 6*Boole[5<=k<=n-5] +(n-k+1)*Boole[n-4<=k<=n]]];
A168646[n_, k_]:= Binomial[n,k]*f[n,k];
Table[A168646[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 05 2025 *)

T(n,k) := if k = 0 or k = n then 1 else (if n <= 6 then (8 - n)*binomial(n, k) else ratcoef(6*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 4), x, k))$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */

def f(n,k):
    if k==0 or k==n: return 1
    elif 0n-5)
def A168646(n,k): return binomial(n,k)*f(n,k)
print(flatten([[A168646(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 05 2025

T(n, n-k) = T(n, k). - G. C. Greubel, Apr 05 2025

Edited by Franck Maminirina Ramaharo, Jan 02 2019

Data values T(7,3), T(7,4), T(8,4) corrected by G. C. Greubel, Apr 05 2025

A168437 a(n) = 3 + 10*floor(n/2).

Original entry on oeis.org

3, 13, 13, 23, 23, 33, 33, 43, 43, 53, 53, 63, 63, 73, 73, 83, 83, 93, 93, 103, 103, 113, 113, 123, 123, 133, 133, 143, 143, 153, 153, 163, 163, 173, 173, 183, 183, 193, 193, 203, 203, 213, 213, 223, 223, 233, 233, 243, 243, 253, 253, 263, 263, 273, 273, 283

Offset: 1

Vincenzo Librandi, Nov 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Bisections of A168437 are A017305 and (A017305 MINUS {3}). - Rick L. Shepherd, Jun 17 2010

[3+10*Floor(n/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013

Table[3 + 10 Floor[n/2], {n, 70}] (* or *) CoefficientList[Series[(3 + 10 x - 3 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)
LinearRecurrence[{1,1,-1},{3,13,13},70] (* Harvey P. Dale, May 26 2021 *)

a(n)=n\2*10+3 \\ Charles R Greathouse IV, Jan 11 2012

a(n) = 10*n - a(n-1) - 4, with n>1, a(1) = 3.
a(n) = 10*floor(n/2) + 3 = A168641(n) + 3. - Rick L. Shepherd, Jun 17 2010
G.f.: x*(3 + 10*x - 3*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 19 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 19 2013
a(n) = (1 + 5*(-1)^n + 10*n)/2. - Bruno Berselli, Sep 19 2013
E.g.f.: (1/2)*(5 - 6*exp(x) + (10*x + 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 22 2016

Edited by Rick L. Shepherd, Jun 17 2010

Definition rewritten, using Shepherd's formula, by Vincenzo Librandi, Sep 19 2013

cp's OEIS Frontend

A168643 Triangle read by rows: T(n,k) = [x^k] p(x,n) where p(x,0) = 1, p(x,n) = (6 - n)*(1+x)^n - (5-n)*(1 + x^n) for 1 <= n <= 4, and p (x,n) = 4*(1+x)^n - Sum_{i=0..2} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) for n >= 5.

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A168644 Triangle read by rows: T(n, k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (7 - n)*(1+x)^n - (6-n)*(1 + x^n) for 1 <= n <= 5, and p(x,n) = 5*(1 + x)^n - Sum_{i=0..3} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) + (1/6)*n*(n - 1)*(n - 5)*x^(n-3) for n >= 6.

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Maxima

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A168646 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (8 - n)*(1+x)^n - (7 - n)*(1 + x^n) for 1 <= n <= 6, and p(x,n) = 6*(1+x)^n - Sum_{i=0..4} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) for n >= 7.

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Examples

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Crossrefs

Programs

Mathematica

Maxima

SageMath

Formula

Extensions

A168437 a(n) = 3 + 10*floor(n/2).

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Mathematica

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A168643 Triangle read by rows: T(n,k) = [x^k] p(x,n) where p(x,0) = 1, p(x,n) = (6 - n)(1+x)^n - (5-n)(1 + x^n) for 1 <= n <= 4, and p (x,n) = 4(1+x)^n - Sum_{i=0..2} (Sum_{j=0..i} binomial(n, j)(x^j + x^(n-j))) for n >= 5.

A168644 Triangle read by rows: T(n, k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (7 - n)(1+x)^n - (6-n)(1 + x^n) for 1 <= n <= 5, and p(x,n) = 5(1 + x)^n - Sum_{i=0..3} (Sum_{j=0..i} binomial(n, j)(x^j + x^(n-j))) + (1/6)n(n - 1)(n - 5)x^(n-3) for n >= 6.

A168646 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (8 - n)(1+x)^n - (7 - n)(1 + x^n) for 1 <= n <= 6, and p(x,n) = 6(1+x)^n - Sum_{i=0..4} (Sum_{j=0..i} binomial(n, j)(x^j + x^(n-j))) for n >= 7.