A132047 -id:A132047 - cp's OEIS frontend

A144706 Central coefficients of the triangle A132047.

1, 6, 18, 60, 210, 756, 2772, 10296, 38610, 145860, 554268, 2116296, 8112468, 31201800, 120349800, 465352560, 1803241170, 7000818660, 27225405900, 106035791400, 413539586460, 1614773623320, 6312296891160, 24700292182800, 96742811049300, 379231819313256

Offset: 0

Paul Barry, Sep 19 2008

Hankel transform is A144708.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A010686, A039599, A082505, A106566, A132047, A144708.

[n eq 0 select 1 else 3*(n+1)*Catalan(n): n in [0..40]]; // G. C. Greubel, Jun 16 2022

Table[3*Binomial[2n,n] -2*Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 16 2022 *)

a(n) = if(n, 3*binomial(2*n, n), 1) \\ Charles R Greathouse IV, Oct 23 2023

[3*binomial(2*n, n) -2*bool(n==0) for n in (0..40)] # G. C. Greubel, Jun 16 2022

G.f.: 3/sqrt(1-4*x) - 2;
a(n) = 3*binomial(2*n, n) - 2*0^n.
From Philippe Deléham, Oct 30 2008: (Start)
a(n) = Sum_{k=0..n} A039599(n,k)*A010686(k).
a(n) = Sum_{k=0..n} A106566(n,k)*A082505(k+1). (End)
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1). - R. J. Mathar, Nov 30 2012
E.g.f.: -2 + 3*exp(2*x)*BesselI(0, 2*x). - G. C. Greubel, Jun 16 2022

A144707 Diagonal sums of the triangle A132047.

Original entry on oeis.org

1, 1, 2, 7, 11, 22, 35, 61, 98, 163, 263, 430, 695, 1129, 1826, 2959, 4787, 7750, 12539, 20293, 32834, 53131, 85967, 139102, 225071, 364177, 589250, 953431, 1542683, 2496118, 4038803, 6534925, 10573730, 17108659, 27682391, 44791054, 72473447, 117264505

Offset: 0

Paul Barry, Sep 19 2008

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A000045.

Table[3*Fibonacci[n+1] -3 -(-1)^n +2*Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 16 2022 *)

Vec((1-x^2+4*x^3+2*x^4) / ((1-x^2)*(1-x-x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017

[3*fibonacci(n+1) -2 -2*((n+1)%2) +2*bool(n==0) for n in (0..40)] # G. C. Greubel, Jun 16 2022

G.f.: (1 - x^2 + 4*x^3 + 2*x^4) / ((1 - x^2)*(1 - x - x^2)).
a(n) = 3*Fibonacci(n+1) - 3 - (-1)^n + 2*0^n.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>4. - Philippe Deléham, Dec 16 2008
From Colin Barker, Jul 12 2017: (Start)
a(n) = (3*2^(-n-1)*((1 + sqrt(5))^(n+1) - (1-sqrt(5))^(n+1))) / sqrt(5) - 4 for n>0 and even.
a(n) = (3*2^(-n-1)*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1)))/sqrt(5) - 2 for n odd.
(End)

A131128 Binomial transform of [1, 1, 5, 1, 5, 1, 5, ...].

Original entry on oeis.org

1, 2, 8, 20, 44, 92, 188, 380, 764, 1532, 3068, 6140, 12284, 24572, 49148, 98300, 196604, 393212, 786428, 1572860, 3145724, 6291452, 12582908, 25165820, 50331644, 100663292, 201326588, 402653180, 805306364, 1610612732, 3221225468

Offset: 0

Gary W. Adamson, Jun 16 2007

Row sums of triangle A131129. - Emeric Deutsch, Jun 19 2007

For n >= 4, a(n) is the number of vertices in the dendrimer nanostar NS1[n-3] defined pictorially in the Ashrafi et al. reference (Ns1[3] is shown in Fig. 1) or in the Ahmadi et al. reference (Fig. 1). - Emeric Deutsch, May 17 2018

			a(3) = 20 = (1, 3, 3, 1) dot (1, 1, 5, 1) = (1 + 3 + 15 + 1).

B. Monjardet, Acyclic domains of linear orders: a survey, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 139-160. [From N. J. A. Sloane, Feb 07 2009]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. B. Ahmadi and M. Sadeghimehr, Atom bond connectivity index of an infinite class NS1[n] of dendrimer nanostars, Optoelectronics and Advanced Materials, 4(7):1040-1042 July 2010.
Ali Reza Ashrafi and Parisa Nikzad, Kekulé index and bounds of energy for nanostar dendrimers, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A131129, A095121, A132047.

Concatenation([1],List([1..30], n->3*2^n-4)); # Muniru A Asiru, May 17 2018

1, seq(3*2^n-4, n = 1 .. 30); # Emeric Deutsch, Jun 19 2007

CoefficientList[Series[(1-x+4x^2)/((1-x)(1-2x)),{x,0,40}],x] (* Vincenzo Librandi, Apr 11 2012 *)

a(n) = 3*2^n - 4 for n >= 1; a(0)=1. Formula follows by replacing [1,1,5,1,5,1,...] with [1,3-2,3+2,3-2,3+2,3-2,...]. - Emeric Deutsch, Jun 19 2007
G.f.: (1 - x + 4x^2)/((1-x)(1-2x)). - Emeric Deutsch, Jul 09 2007
Row sums of triangle A132047. - Gary W. Adamson, Aug 08 2007
a(n) = 2*a(n-1) + 4 for n >= 2, a(0)=1, a(1)=2. - Philippe Deléham, Sep 23 2009
a(n) = 2*A033484(n-1) for n>0. - R. J. Mathar, Feb 27 2019

A168623 Triangle read by rows: T(n, k) = [x^k]( 9(1+x)^n - 8(1 + x^n) ), with T(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 18, 1, 1, 27, 27, 1, 1, 36, 54, 36, 1, 1, 45, 90, 90, 45, 1, 1, 54, 135, 180, 135, 54, 1, 1, 63, 189, 315, 315, 189, 63, 1, 1, 72, 252, 504, 630, 504, 252, 72, 1, 1, 81, 324, 756, 1134, 1134, 756, 324, 81, 1, 1, 90, 405, 1080, 1890, 2268, 1890, 1080, 405, 90, 1

Offset: 0

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

			Triangle begins as:
  1;
  1,  1;
  1, 18,   1;
  1, 27,  27,    1;
  1, 36,  54,   36,    1;
  1, 45,  90,   90,   45,    1;
  1, 54, 135,  180,  135,   54,    1;
  1, 63, 189,  315,  315,  189,   63,    1;
  1, 72, 252,  504,  630,  504,  252,   72,   1;
  1, 81, 324,  756, 1134, 1134,  756,  324,  81,  1;
  1, 90, 405, 1080, 1890, 2268, 1890, 1080, 405, 90, 1;

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

Cf. A000045, A132047.

A168623:= func< n,k | k eq 0 or k eq n select 1 else 9*Binomial(n,k) >;
[A168623(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 09 2025

(* First program *)
p[x_, n_]:= With[{m=4}, If[n==0, 1, (2*m+1)(1+x)^n - 2*m*(1+x^n)]];
Table[CoefficientList[p[x,n], x], {n,0,10}]//Flatten
(* Second program *)
A168623[n_, k_]:= If[k==0 || k==n, 1, 9*Binomial[n,k]];
Table[A168623[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 09 2025 *)

def A168623(n,k):
    if (k==0 or k==n): return 1
    else: return 9*binomial(n,k)
print(flatten([[A168623(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 09 2025

From G. C. Greubel, Apr 09 2025: (Start)
T(n, k) = binomial(n,k)*( [k=0] + 9*[0 < k < n] + [k=n] ).
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = 9*2^n - 16 + 8*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = -8*(1 + (-1)^n) + 17*[n=0].
Sum_{k=0..floor(n/2)} T(n-k, k) = 9*A000045(n+1) - 4*(3 + (-1)^n) + 8*[n=0]. (End)

Keyword:tabl and row sum formula added - The Assoc. Editors of the OEIS, Dec 05 2009

A168622 Triangle read by rows: T(n, k) = [x^k]( 7(1+x)^n - 6(1+x^n) ) with T(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 21, 21, 1, 1, 28, 42, 28, 1, 1, 35, 70, 70, 35, 1, 1, 42, 105, 140, 105, 42, 1, 1, 49, 147, 245, 245, 147, 49, 1, 1, 56, 196, 392, 490, 392, 196, 56, 1, 1, 63, 252, 588, 882, 882, 588, 252, 63, 1, 1, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1

Offset: 0

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

			Triangle begins as:
  1;
  1,  1;
  1, 14,   1;
  1, 21,  21,   1;
  1, 28,  42,  28,    1;
  1, 35,  70,  70,   35,    1;
  1, 42, 105, 140,  105,   42,    1;
  1, 49, 147, 245,  245,  147,   49,   1;
  1, 56, 196, 392,  490,  392,  196,  56,   1;
  1, 63, 252, 588,  882,  882,  588, 252,  63,  1;
  1, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1;

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

Cf. A022090, A131115, A132047, A168620, A168623.

Columns (essentially): A008589 (k=1), A024966 (k=2).

A168622:= func< n,k | k eq 0 or k eq n select 1 else 7*Binomial(n,k) >;
[A168622(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2025

(* First program *)
p[x_, n_]:= With[{m=3}, If[n==0, 1, (2*m+1)(1+x)^n - 2*m*(1+x^n)]];
Table[CoefficientList[p[x,n], x], {n,0,12}]//Flatten
(* Second program *)
A168622[n_, k_]:= If[k==0 || k==n, 1, 7*Binomial[n,k]];
Table[A168622[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 10 2025 *)

def A168622(n,k):
    if k==0 or k==n: return 1
    else: return 7*binomial(n,k)
print(flatten([[A168622(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 10 2025

From G. C. Greubel, Apr 10 2025: (Start)
T(n, k) = 7*binomial(n, k), with T(n, 0) = T(n, n) = 1.
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = 2*A048489(n-1) + 6*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = -6*(1 + (-1)^n) + 13*[n=0].
Sum_{k=0..floor(n/2)} T(n-k, k) = A022090(n+1) - 3*(3 + (-1)^n) + 6*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (14/sqrt(3))*(-1)^n*cos((4*n+1)*Pi/6) - 6*(1 + (-1)^n*cos(n*Pi/2)) + 6*[n=0]. (End)

A168621 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,k) = ((n - 1)! + 1)*binomial(n, k) for 1 <= k <= n - 1, n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 28, 42, 28, 1, 1, 125, 250, 250, 125, 1, 1, 726, 1815, 2420, 1815, 726, 1, 1, 5047, 15141, 25235, 25235, 15141, 5047, 1, 1, 40328, 141148, 282296, 352870, 282296, 141148, 40328, 1, 1, 362889, 1451556, 3386964, 5080446, 5080446, 3386964, 1451556, 362889, 1

Offset: 0

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

Row 0 is 1, and row n gives the coefficients in the expansion of (x + 1)^n + (n - 1)!*((x + 1)^n - x^n -1). - Franck Maminirina Ramaharo, Dec 22 2018

			Triangle begins:
  1;
  1,     1;
  1,     4,      1;
  1,     9,      9,      1;
  1,    28,     42,     28,      1;
  1,   125,    250,    250,    125,      1;
  1,   726,   1815,   2420,   1815,    726,      1;
  1,  5047,  15141,  25235,  25235,  15141,   5047,     1;
  1, 40328, 141148, 282296, 352870, 282296, 141148, 40328, 1;
  ...

Row sums: A154252.

Cf. A007318, A132047, A155863, A155864, A155865 A168619, A219570.

p[x_, n_] := If[n == 0, 1, (x + 1)^n + (n - 1)!*((x + 1)^n - x^n - 1)];
Table[CoefficientList[p[x, n], x], {n, 0, 12}] // Flatten (* Franck Maminirina Ramaharo, Dec 22 2018 *)

T(n, k) := if k = 0 or k = n then 1 else ((n - 1)! + 1)*binomial(n, k)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 22 2018 */

From Franck Maminirina Ramaharo, Dec 22 2018: (Start)
T(n,k) = A007318(n,k) + A219570(n,k) for 1 <= k <= n - 1, n >= 2.
E.g.f.: exp((1 + x)*y) + log((1 - y)*(1 - x*y)/(1 - (1 + x)*y)). (End)

Edited by Franck Maminirina Ramaharo, Dec 22 2018 (previous definition and examples were the same as A168620, but with different entries, as pointed out by R. J. Mathar, Oct 21 2012)

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A144706 Central coefficients of the triangle A132047.

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A144707 Diagonal sums of the triangle A132047.

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A131128 Binomial transform of [1, 1, 5, 1, 5, 1, 5, ...].

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A168623 Triangle read by rows: T(n, k) = [x^k]( 9*(1+x)^n - 8*(1 + x^n) ), with T(0, 0) = 1.

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A168622 Triangle read by rows: T(n, k) = [x^k]( 7*(1+x)^n - 6*(1+x^n) ) with T(0, 0) = 1.

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A168621 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,k) = ((n - 1)! + 1)*binomial(n, k) for 1 <= k <= n - 1, n >= 2.

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A168623 Triangle read by rows: T(n, k) = [x^k]( 9(1+x)^n - 8(1 + x^n) ), with T(0, 0) = 1.

A168622 Triangle read by rows: T(n, k) = [x^k]( 7(1+x)^n - 6(1+x^n) ) with T(0, 0) = 1.