A118414 -id:A118414 - cp's OEIS frontend

A118413 Triangle read by rows: T(n,k) = (2n-1)2^(k-1), 0

1, 3, 6, 5, 10, 20, 7, 14, 28, 56, 9, 18, 36, 72, 144, 11, 22, 44, 88, 176, 352, 13, 26, 52, 104, 208, 416, 832, 15, 30, 60, 120, 240, 480, 960, 1920, 17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 21, 42, 84, 168

Offset: 1

Reinhard Zumkeller, Apr 27 2006

Central terms give A118415; row sums give A118414;
T(n,1) = A005408(n-1);
T(n,2) = A016825(n-1) for n>1;
T(n,3) = A017113(n-1) for n>2;
T(n,4) = A051062(n-1) for n>3;
T(n,n-2) = A052951(n-1) for n>2;
T(n,n) = A014480(n-1) = A118416(n,n);
A001511(T(n,k)) = A002260(n,k);
A003602(T(n,k)) = A002024(n,k).
G.f.: x*y*(1 + x + 2*x*y - 6*x^2*y)/((1 - x)^2*(1 - 2*x*y)^2). - Stefano Spezia, Dec 22 2024

			   1
   3   6
   5  10  20
   7  14  28  56
   9  18  36  72 144
  11  22  44  88 176 352
  13  26  52 104 208 416  832
  15  30  60 120 240 480  960 1920
  17  34  68 136 272 544 1088 2176 4352
  19  38  76 152 304 608 1216 2432 4864 9728
  ...

Cf. A001511, A002024, A002260, A003602, A005408, A014480, A016825, A017113, A051062, A052951, A117303, A118414, A118415, A118416, A117303.

Select[Flatten[Table[(2n-1)2^(k-1),{n,20},{k,0,n}]],IntegerQ] (* Harvey P. Dale, Jan 17 2024 *)

from math import isqrt, comb
def A118413(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    return ((a<<1)-1)<Chai Wah Wu, Jun 20 2025

A014477 Expansion of (1 + 2x)/(1 - 2x)^3.

Original entry on oeis.org

1, 8, 36, 128, 400, 1152, 3136, 8192, 20736, 51200, 123904, 294912, 692224, 1605632, 3686400, 8388608, 18939904, 42467328, 94633984, 209715200, 462422016, 1015021568, 2218786816, 4831838208, 10485760000, 22682796032, 48922361856, 105226698752, 225754218496

Offset: 0

N. J. A. Sloane

The sequence 0,1,8,... has a(n) = n^2*2^(n-1) and is the binomial transform of the hexagonal numbers A000384 (with leading 0). - Paul Barry, Jun 09 2003
As 0,1,8,... this is n^2*2^(n-1), the binomial transform of the hexagonal numbers A000384 (include the leading 0). Partial sums are A036826. - Paul Barry, Jun 10 2003
Sequence gives total value of all possible sums of distinct odd integers with maximum term less than 2n+1. E.g., for a(3) we can have the sums 1, 3, 5, 1+3, 1+5, 3+5, 1+3+5, which sum to 1+3+5+4+6+8+9 = 36. - Jon Perry, Feb 06 2004
Number of edges on a partially truncated (n+1)-cube (column 2 of A271316).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A007758, A036826, A118414, A355234.

[(n+1)^2*2^n: n in [0..35]]; // Vincenzo Librandi, Aug 21 2011

a:=n->sum(binomial(n,j)*n*j,j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Oct 19 2006
a:=n->sum(n*numbcomb(n)/2, j=1..n): seq(a(n), n=1..25); # Zerinvary Lajos, Apr 25 2007

f[n_]:=(n^2*2^n)/2;Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2009 *)

a(n)=(n+1)^2*2^n \\ Charles R Greathouse IV, Apr 07 2016

O.g.f.: (1 + 2*x)/(1 - 2*x)^3 (see the name).
a(n) = (n+1)^2*2^n = A007758(n+1)/2. - Henry Bottomley, Jun 13 2001
The binomial transform of 0, 1, 8, ... is A077616. - Paul Barry, Jul 24 2003
a(1)=1, a(n) = 2a(n-1) + (2n-1)*2^(n-1). - Jon Perry, Feb 06 2004
a(n) = sum of (n+1)-th row of the triangle in A118416. - Reinhard Zumkeller, Apr 27 2006
a(n) = Sum_{j=0..n} binomial(n,j)*n*j. - Zerinvary Lajos, Oct 19 2006
E.g.f.: exp(2*x)*(1 + 6*x + 8*x^2/2!). - Wolfdieter Lang, Jul 29 2017
Sum_{n>=0} 1/a(n) = Pi^2/6 - log(2)^2. - Daniel Suteu, Oct 31 2017
Sum_{n>=0} (-1)^n/a(n) = -2 *  Li_2(-1/2) = -2 * A355234. - Amiram Eldar, Oct 01 2022

A277046 Triangle read by rows: T(n,k) = 2^n - n + k - 1 for n >= 1, with 1 <= k <= 2n-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 27, 28, 29, 30, 31, 32, 33, 34, 35, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519

Offset: 1

Miquel Cerda, Sep 27 2016

			Triangle begins:
1;
2,   3,  4;
5,   6,  7,  8,  9;
12, 13, 14, 15, 16, 17, 18;
27, 28, 29, 30, 31, 32, 33, 34, 35;
58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68;
...
Written as an isosceles triangle the sequence begins:
.                      1;
.                  2,  3,  4;
.              5,  6,  7,  8,  9;
.         12, 13, 14, 15, 16, 17, 18;
.     27, 28, 29, 30, 31, 32, 33, 34, 35;
. 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68;
..

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Row lengths are A005408.
Row sums give A118414.
Column 1 gives A000325, n>=1.
Middle diagonal gives A000225.
Right border gives A083706.
Cf. A118413.

Table[2^n-n+k-1,{n,10},{k,2n-1}]//Flatten (* Harvey P. Dale, Nov 27 2021 *)

Definition from Omar E. Pol, Sep 28 2016

cp's OEIS Frontend

A118413 Triangle read by rows: T(n,k) = (2n-1)2^(k-1), 0

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A014477 Expansion of (1 + 2x)/(1 - 2x)^3.

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A277046 Triangle read by rows: T(n,k) = 2^n - n + k - 1 for n >= 1, with 1 <= k <= 2n-1.

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cp's OEIS Frontend

A118413 Triangle read by rows: T(n,k) = (2*n-1)*2^(k-1), 0

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A014477 Expansion of (1 + 2*x)/(1 - 2*x)^3.

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Magma

Maple

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Formula

A277046 Triangle read by rows: T(n,k) = 2^n - n + k - 1 for n >= 1, with 1 <= k <= 2n-1.

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Extensions

A118413 Triangle read by rows: T(n,k) = (2n-1)2^(k-1), 0

A014477 Expansion of (1 + 2x)/(1 - 2x)^3.