A014477 -id:A014477 - cp's OEIS frontend

A007758 a(n) = 2^n*n^2.

0, 2, 16, 72, 256, 800, 2304, 6272, 16384, 41472, 102400, 247808, 589824, 1384448, 3211264, 7372800, 16777216, 37879808, 84934656, 189267968, 419430400, 924844032, 2030043136, 4437573632, 9663676416, 20971520000, 45365592064, 97844723712, 210453397504

Offset: 0

David J. Snook (ua532(AT)freenet.victoria.bc.ca)

"The traveling salesman problem can be solved in time O(n^2 2^n) (where n is the size of the network to visit)." [Wikipedia] - Jonathan Vos Post, Apr 10 2006

Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - N. J. A. Sloane, Feb 08 2017

Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
Wikipedia, Complexity.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
Index entries for sequences related to Benford's law.

Cf. A014477, A248917, A355234.

[2^n*n^2: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n*n^k, k=2..2), n=0..25); and seq(2^n*n^2, n=0..25); # Zerinvary Lajos, Jul 01 2007

Table[n^2 * 2^n, {n, 0, 31}] (* Alonso del Arte, Oct 22 2014 *)
LinearRecurrence[{6,-12,8},{0,2,16},30] (* Harvey P. Dale, Jan 27 2017 *)

a(n)=n^2<Charles R Greathouse IV, Oct 28 2014

From Henry Bottomley, Jun 13 2001: (Start)
a(n) = 2*A014477(n-1).
G.f.: 2*x(1+2*x)/(1-2*x)^3.
Binomial transform of A002939.
Inverse binomial transform of A062189. (End)
Sum_{n>=1} 1/a(n) = Pi^2/12 - (1/2)*(log(2))^2. - Benoit Cloitre, Apr 05 2002
a(n) = Sum_{k=1..n} k*2^k. - Zerinvary Lajos, Oct 09 2006
E.g.f.: exp(2*x)*(2*x + 4*x^2). - Geoffrey Critzer, Aug 28 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = -Li_2(-1/2) (A355234). - Amiram Eldar, Jun 28 2022

A118416 Triangle read by rows: T(n,k) = (2k-1)2^(n-1), 0 < k <= n.

Original entry on oeis.org

1, 2, 6, 4, 12, 20, 8, 24, 40, 56, 16, 48, 80, 112, 144, 32, 96, 160, 224, 288, 352, 64, 192, 320, 448, 576, 704, 832, 128, 384, 640, 896, 1152, 1408, 1664, 1920, 256, 768, 1280, 1792, 2304, 2816, 3328, 3840, 4352, 512, 1536, 2560, 3584, 4608, 5632, 6656, 7680

Offset: 1

Reinhard Zumkeller, Apr 27 2006

Row sums give A014477: Sum_{k=1..n} T(n,k) = A014477(n-1);
central terms give A118415; T(2*k-1,k) = A058962(k-1);
T(n,1) = A000079(n-1);
T(n,2) = A007283(n-1) for n > 1;
T(n,3) = A020714(n-1) for n > 2;
T(n,4) = A005009(n-1) for n > 3;
T(n,5) = A005010(n-1) for n > 4;
T(n,n-1) = A118417(n-1) for n > 1;
T(n,n) = A014480(n-1) = A118413(n,n);
A001511(T(n,k)) = A002024(n,k);
A003602(T(n,k)) = A002260(n,k).
The alternating row sums, Sum_{k=1..n} (-1)^(k+1)*T(n,k), are: (a) in odd rows, the central term, T(n,(n+1)/2) = A058962((n-1)/2); (b) in even rows, the negation of the average of the two central terms, -(T(2n,n) + T(2n,+1))/2 = -A018215(m/2). The absolute values of the alternating row sums give the plain row means, Sum_{k=1..n} T(n,k)/n; the alternating sign row means are (-2)^(n-1). - Gregory Gerard Wojnar, Feb 10 2024

			Triangle begins:
   1;
   2,   6;
   4,  12,  20;
   8,  24,  40,  56;
  16,  48,  80, 112, 144;
  32,  96, 160, 224, 288, 352;
  64, 192, 320, 448, 576, 704, 832;

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A118413, A117303, A054582.

a118416 n k = a118416_tabl !! (n-1) !! (k-1)
a118416_row 1 = [1]
a118416_row n = (map (* 2) $ a118416_row (n-1)) ++ [a014480 (n-1)]
a118416_tabl = map a118416_row [1..]
-- Reinhard Zumkeller, Jan 22 2012

A118416 := proc(n,k) 2^(n-1)*(2*k-1) ; end proc: # R. J. Mathar, Sep 04 2011

Flatten[Table[(2k-1)2^(n-1),{n,10},{k,n}]] (* Harvey P. Dale, Aug 26 2014 *)

from math import isqrt
def A118416(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(1-a)+(n<<1)-1<Chai Wah Wu, Jun 20 2025

T(n,k) = 2*T(n-1,k), 1 <= k < n; T(n,n) = A014480(n-1).

A014479 Exponential generating function = (1+2x)/(1-2x)^3.

Original entry on oeis.org

1, 8, 72, 768, 9600, 138240, 2257920, 41287680, 836075520, 18579456000, 449622835200, 11771943321600, 331576403558400, 9998303861145600, 321374052679680000, 10969567664799744000, 396275631890890752000

Offset: 0

N. J. A. Sloane

nonn

Robert Israel, Table of n, a(n) for n = 0..395
Ben Adenbaum, Jennifer Elder, Pamela E. Harris, and J. Carlos Martínez Mori, Boolean intervals in the weak Bruhat order of a finite Coxeter group, arXiv:2403.07989 [math.CO], 2024. See p. 8.

Cf. A001620, A014477, A187735.

seq(add(count(Composition(k))*count(Permutation(k)),k=1..n),n=1..17); # Zerinvary Lajos, Oct 17 2006
seq(2^n*(n+1)^2*n!, n=0..30); # Robert Israel, Oct 28 2015

Table[2^n (n+1)^2 n!, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)

{a(n)=polcoeff( sum(m=0,n,(2*m+1)^(m+1)*x^m / (1 + (2*m+1)*x +x*O(x^n))^(m+1)),n)} \\ Paul D. Hanna, Jan 02 2013
for(n=0,20,print1(a(n),", "))

vector(30, n, n--; n!*(n+1)^2*2^n) \\ Altug Alkan, Oct 28 2015

a(n) = A014477(n) * n!. - Franklin T. Adams-Watters, Nov 02 2006
G.f.: Sum_{n>=0} (2*n+1)^(n+1) * x^n / (1 + (2*n+1)*x)^(n+1). - Paul D. Hanna, Jan 02 2013
From Vladimir Reshetnikov, Oct 28 2015: (Start)
a(n) = 2^n*(n+1)^2*n!.
Recurrence: a(0) = 1, n*a(n) = 2*(n+1)^2*a(n-1). (End)
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = 2*(Ei(1/2) - gamma + log(2)), where Ei(x) is the exponential integral and gamma is Euler's constant (A001620).
Sum_{n>=0} (-1)^n/a(n) = 2*(gamma - Ei(-1/2) - gamma - log(2)). (End)

A036826 a(n) = A036800(n)/2.

Original entry on oeis.org

0, 1, 9, 45, 173, 573, 1725, 4861, 13053, 33789, 84989, 208893, 503805, 1196029, 2801661, 6488061, 14876669, 33816573, 76283901, 170917885, 380633085, 843055101, 1858076669, 4076863485, 8908701693, 19394461693, 42077257725, 90999619581, 196226318333

Offset: 0

N. J. A. Sloane

Binomial transform of A054569 (with leading 0). Partial sums of A014477 (with leading 0). - Paul Barry, Jun 11 2003

This sequence is related to A000337 by a(n) = n*A000337(n) - Sum_{i=0..n-1} A000337(i). - Bruno Berselli, Mar 06 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A000337, A014477, A036800, A054569, A209359.

m:=28; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1+2*x)/((1-x)*(1-2*x)^3))); // Bruno Berselli, Mar 06 2012

A036826:= n-> 2^n*(3-2*n+n^2) -3; seq(A036826(n), n=0..30); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{7,-18,20,-8}, {0,1,9,45}, 29] (* Bruno Berselli, Mar 06 2012 *)

for(n=0, 28, print1(2^n*(n^2-2*n+3)-3", ")); \\ Bruno Berselli, Mar 06 2012

[2^n*(3-2*n+n^2) -3 for n in (0..30)] # G. C. Greubel, Mar 31 2021

From Paul Barry, Jun 11 2003: (Start)
G.f.: x*(1+2*x)/((1-x)*(1-2*x)^3).
a(n) = 2^n*(n^2-2*n+3) - 3.
a(n) = Sum_{k=0..n} k^2*2^(k-1). (End)
a(n) = 7*a(n-1) -18*a(n-2) +20*a(n-3) -8*a(n-4). - Harvey P. Dale, Mar 04 2015
E.g.f.: -3*exp(x) + (3 -2*x +4*x^2)*exp(2*x). - G. C. Greubel, Mar 31 2021

A118414 a(n) = (2n - 1) (2^n - 1).

Original entry on oeis.org

1, 9, 35, 105, 279, 693, 1651, 3825, 8687, 19437, 42987, 94185, 204775, 442341, 950243, 2031585, 4325343, 9175005, 19398619, 40894425, 85983191, 180355029, 377487315, 788529105, 1644167119, 3422552013, 7113539531, 14763950025, 30601641927, 63350767557, 130996502467, 270582939585

Offset: 1

Reinhard Zumkeller, Apr 27 2006

Row sums of triangle A118413.
For fixed n, define a triangle T(r,c) counting down the first n odd numbers on the left side, T(r,1) = 2*(n-r)+1, and counting up odd numbers on the right side, T(r,r) = 2*(n+r)-3, r>1. The interior elements are set by T(r,c)=T(r-1,c-1) + T(r-1,c). The sum of all members in this triangle is a(n). - J. M. Bergot, Oct 12 2012
Row sums of triangle A277046. - Miquel Cerda, Sep 28 2016

			The triangle T(r,c) for n=4 has row(1)=7; row(2) = 5, 9; row(3) = 3, 14, 11; row(4) = 1, 17, 25, 13, and a sum of 7+5+9+...+13 = 105 = a(4). - _J. M. Bergot_, Oct 12 2012

Altug Alkan, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4)

Cf. A014477, A000225, A005408, A118413, A277046.

[(2*n-1)*(2^n-1): n in [1..40]]; // Vincenzo Librandi, Dec 26 2010

Table[(2 n - 1) (2^n - 1), {n, 32}] (* or *)
Rest@ CoefficientList[Series[-x (-1 - 3 x + 6 x^2)/((2 x - 1)^2*(x - 1)^2), {x, 0, 32}], x] (* Michael De Vlieger, Sep 26 2016 *)
LinearRecurrence[{6,-13,12,-4},{1,9,35,105},40] (* Harvey P. Dale, Sep 12 2023 *)

a(n)=(2*n-1)*(2^n-1) \\ Charles R Greathouse IV, Oct 12 2012

a(n) = A005408(n-1)*(A000079(n) - 1). Corrected by Omar E. Pol, Sep 26 2016
G.f. -x*(-1-3*x+6*x^2) / ( (2*x-1)^2*(x-1)^2 ). - R. J. Mathar, Oct 15 2012
a(n) = A005408(n-1)*A000225(n). - Miquel Cerda, Sep 26 2016

A086603 a(n) = n^3*3^(n-1).

Original entry on oeis.org

0, 1, 24, 243, 1728, 10125, 52488, 250047, 1119744, 4782969, 19683000, 78594219, 306110016, 1167575877, 4374822312, 16142520375, 58773123072, 211488540273, 753145430616, 2657317134051, 9298091736000, 32291110337661

Offset: 0

Paul Barry, Jul 23 2003

Binomial transform of A086604. Second binomial transform of A086605.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

Cf. A014477, A086604, A086605.

List([0..30], n-> 3^(n-1)*n^3 ); # G. C. Greubel, Feb 08 2020

[3^(n-1)*n^3: n in [0..30]]; // G. C. Greubel, Feb 08 2020

seq( 3^(n-1)*n^3, n=0..30); # G. C. Greubel, Feb 08 2020

Table[n^3 3^(n-1),{n,0,30}]  (* Harvey P. Dale, Mar 12 2011 *)

vector(31, n, my(m=n-1); 3^(m-1)*m^3) \\ G. C. Greubel, Feb 08 2020

[3^(n-1)*n^3 for n in (0..30)] # G. C. Greubel, Feb 08 2020

From G. C. Greubel, Feb 08 2020: (Start)
G.f.: x*(1 + 12*x + 9*x^2)/(1-3*x)^4.
E.g.f.: x*(1 + 9*x + 9*x^2)*exp(x). (End)

A141692 Triangle read by rows: T(n,k) = n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 0 <= k <= n.

Original entry on oeis.org

0, -1, 1, -2, 0, 2, -3, -3, 3, 3, -4, -8, 0, 8, 4, -5, -15, -10, 10, 15, 5, -6, -24, -30, 0, 30, 24, 6, -7, -35, -63, -35, 35, 63, 35, 7, -8, -48, -112, -112, 0, 112, 112, 48, 8, -9, -63, -180, -252, -126, 126, 252, 180, 63, 9, -10, -80, -270, -480, -420, 0, 420, 480, 270, 80, 10

Offset: 0

Roger L. Bagula, Sep 09 2008

The row sums are zero.

Row n consists of the coefficients in the expansion of n*(x - 1)*(x + 1)^(n - 1). - Franck Maminirina Ramaharo, Oct 02 2018

			Triangle begins:
    0;
   -1,   1;
   -2,   0,    2;
   -3,  -3,    3,    3;
   -4,  -8,    0,    8,    4;
   -5, -15,  -10,   10,   15,   5;
   -6, -24,  -30,    0,   30,  24,   6;
   -7, -35,  -63,  -35,   35,  63,  35,   7;
   -8, -48, -112, -112,    0, 112, 112,  48,   8;
   -9, -63, -180, -252, -126, 126, 252, 180,  63,  9;
  -10, -80, -270, -480, -420,   0, 420, 480, 270, 80, 10;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..5150 (Rows n=1..100 of triangle, flattened; offset adapted by _Georg Fischer_, Jan 31 2019)
Rida T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
Eric Weisstein's World of Mathematics, Bernstein Polynomial
Wikipedia, Bernstein polynomial

Cf. A007318, A112467, A128433, A128434.

a:=proc(n,k) n*(binomial(n-1,k-1)-binomial(n-1,k)); end proc: seq(seq(a(n,k),k=0..n),n=0..10); # Muniru A Asiru, Oct 03 2018

Table[Table[n*(Binomial[n - 1, k - 1] - Binomial[n - 1, k]),{k, 0, n}],{n, 0, 12}]//Flatten

T(n, k) := n*(binomial(n - 1, k - 1) - binomial(n - 1, k))$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 02 2018 */

T(n,k) = n*(B(1/2;n-1,k-1) - B(1/2;n-1,k))*2^(n - 1), where B(t;n,k) = binomial(n,k)*t^k*(1 - t)^(n - k) denotes the k-th Benstein basis polynomial of degree n.
T(n,k) = n*A112467(n,k).
From Franck Maminirina Ramaharo, Oct 02 2018: (Start)
T(n,k) = -T(n,n-k)
T(n,0) = -n.
T(n,1) = -A067998(n)
E.g.f.: (x*y - y)/(x*y + y - 1)^2.
Sum_{k=0..n} abs(T(n,k)) = 2*A100071(n).
Sum_{k=0..n} T(n,k)^2 = 2*A037965(n).
Sum_{k=0..n} k*T(n,k) = A001787(n).
Sum_{k=0..n} k^2*T(n,k) = A014477(n-1). (End)

Edited, new name and offset corrected by Franck Maminirina Ramaharo, Oct 02 2018

A152548 Sum of squared terms in rows of triangle A152547: a(n) = Sum_{k=0..C(n,[n/2])-1} A152547(n,k)^2.

Original entry on oeis.org

1, 4, 10, 24, 54, 120, 260, 560, 1190, 2520, 5292, 11088, 23100, 48048, 99528, 205920, 424710, 875160, 1798940, 3695120, 7574996, 15519504, 31744440, 64899744, 132503644, 270415600, 551231800, 1123264800, 2286646200, 4653525600

Offset: 0

Paul D. Hanna, Dec 14 2008

nonn

Cf. A008288, A152547, A107233, A014477.

seq(simplify((-2)^n*hypergeom([-n,3/2], [1], 2)),n=0..29); # Peter Luschny, Apr 26 2016

CoefficientList[Series[Sqrt[(1+2x)/(1-2x)^3],{x,0,30}],x] (* Harvey P. Dale, Jan 04 2016 *)

a(n)=sum(k=0,floor((n+1)/2),binomial(n+1, k)*(n+1-2*k)^3)/(n+1)

G.f.: A(x) = sqrt( (1+2x)/(1-2x)^3 ).
a(n) = Sum_{k=0..[(n+1)/2]} C(n+1, k)*(n+1-2k)^3/(n+1).
a(n) = A107233(n)/(n+1).
Self-convolution equals A014477.
E.g.f.: ((1 + 4*x)*BesselI(0, 2*x) + 4*x*BesselI(1, 2*x)). - Peter Luschny, Aug 26 2012
a(n) = (-2)^n*hypergeom([-n,3/2], [1], 2). - Peter Luschny, Apr 26 2016
D-finite with recurrence: (n+1)*a(n+1) = 4*a(n) + 4*n*a(n-1). - Vladimir Reshetnikov, Oct 10 2016
a(n) ~ 2^(n + 3/2) * sqrt(n/Pi). - Vaclav Kotesovec, Oct 11 2016
From Peter Bala, Mar 31 2024: (Start)
a(n) = (2^n) * Sum_{k = 0..n} (-1)^(n+k)*binomial(1/2, k)*binomial(-3/2, n-k).
a(n) = (2^n) * Sum_{k = 0..n} (2^k)*binomial(n, k)*binomial(1/2, k).
a(n) = (2^n)* Sum_{k = 0..n} binomial(n, k)*binomial(k+1/2, n). See A008288.
a(n) = (2*n + 1)!/(2^n * n!^2) * hypergeom([-n, -1/2], [-n-1/2], -1).
a(n) = 2^n * hypergeom([-n, -1/2], [1], 2).
a(n) = (-1/2)^n * binomial(2*n, n)/(1 - 2*n) * hypergeom([-n, 3/2], [-n+3/2], -1).(End)

A084850 2^(n-1)*(n^2+2n+2).

Original entry on oeis.org

1, 5, 20, 68, 208, 592, 1600, 4160, 10496, 25856, 62464, 148480, 348160, 806912, 1851392, 4210688, 9502720, 21299200, 47448064, 105119744, 231735296, 508559360, 1111490560, 2420113408, 5251268608, 11358175232, 24494735360

Offset: 0

Paul Barry, Jun 09 2003

Binomial transform of A084849. a(n)=A014477(n-1)+A001787(n+1).

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A134083.

LinearRecurrence[{6,-12,8},{1,5,20},40] (* Harvey P. Dale, Mar 15 2016 *)

G.f.: (1 - x+2x^2)/(1 - 2x)^3.

Equals A134083 * [1,2,3,...]. - Gary W. Adamson, Oct 07 2007

A135065 A127733 * A007318 as infinite lower triangular matrices.

Original entry on oeis.org

1, 4, 4, 9, 18, 9, 16, 48, 48, 16, 25, 100, 150, 100, 25, 36, 180, 360, 360, 180, 36, 49, 294, 735, 980, 735, 294, 49, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 81, 648, 2268, 4536, 5670, 4536, 2268, 648, 81, 100, 900, 3600, 8400, 12600, 12600, 8400, 3600

Offset: 0

Gary W. Adamson, Nov 16 2007

A135065 * [1/1, 1/2, 1/3, ...] = A066524: (1, 6, 21, 60, 155, ...).

Triangle T(n,k), 0 <= k <= n, read by rows, given by (4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 27 2011

			First few rows of the triangle:
   1;
   4,   4;
   9,  18,   9;
  16,  48,  48,  16;
  25, 100, 150, 100,  25;
  36, 180, 360, 360, 180,  36;
  49, 294, 735, 980, 735, 294,  49;

G. C. Greubel, Table of n, a(n) for the first 50 rows
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

Cf. A000290, A127733, A066524, A014477 (row sums), A084938.

with(combstruct):for n from 0 to 11 do seq(n*m*count(Combination(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Apr 09 2008

Flatten[Table[Binomial[n,k](n+1)^2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 12 2013 *)

T(n,k) = binomial(n,k)*(n+1)^2 = A007318(n,k)*A000290(n+1). - Philippe Deléham, Oct 27 2011
T(n-1,k-1) = Sum_{i=-k..k} (-1)^i*(k^2-i^2)*binomial(n,k+i)*binomial(n,k-i). - Mircea Merca, Apr 05 2012
G.f.: (-1 - x - x*y)/(x + x*y - 1)^3. - R. J. Mathar, Aug 12 2015

Corrected by Zerinvary Lajos, Apr 09 2008

cp's OEIS Frontend

A007758 a(n) = 2^n*n^2.

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A118416 Triangle read by rows: T(n,k) = (2*k-1)*2^(n-1), 0 < k <= n.

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A014479 Exponential generating function = (1+2*x)/(1-2*x)^3.

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A036826 a(n) = A036800(n)/2.

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A118414 a(n) = (2*n - 1) * (2^n - 1).

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A086603 a(n) = n^3*3^(n-1).

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GAP

Magma

Maple

Mathematica

A118416 Triangle read by rows: T(n,k) = (2k-1)2^(n-1), 0 < k <= n.

A014479 Exponential generating function = (1+2x)/(1-2x)^3.

A118414 a(n) = (2n - 1) (2^n - 1).