A014480 -id:A014480 - cp's OEIS frontend

A167580 A triangle related to the a(n) formulas of the rows of the ED3 array A167572.

1, 6, -1, 20, 0, 3, 56, 28, 98, -15, 144, 192, 1080, -48, 105, 352, 880, 7568, 2024, 6534, -945, 832, 3328, 40976, 31616, 132444, -8112, 10395, 1920, 11200, 187488, 274480, 1593960, 286900, 972162, -135135, 4352, 34816, 761600, 1784320, 13962848

Offset: 1

Johannes W. Meijer, Nov 10 2009

The a(n) formulas given below correspond to the first ten rows of the ED3 array A167572.

The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

			Row 1: a(n) = 1.
Row 2: a(n) = 6*n - 1.
Row 3: a(n) = 20*n^2 + 0*n + 3.
Row 4: a(n) = 56*n^3 + 28*n^2 + 98*n - 15.
Row 5: a(n) = 144*n^4 + 192*n^3 + 1080*n^2 - 48*n + 105.
Row 6: a(n) = 352*n^5 + 880*n^4 + 7568*n^3 + 2024*n^2 + 6534*n - 945.
Row 7: a(n) = 832*n^6 + 3328*n^5 + 40976*n^4 + 31616*n^3 + 132444*n^2 - 8112*n + 10395.
Row 8: a(n) = 1920*n^7 + 11200*n^6 + 187488*n^5 + 274480*n^4 + 1593960*n^3 + 286900*n^2 + 972162*n - 135135.
Row 9: a(n) = 4352*n^8 + 34816*n^7 + 761600*n^6 + 1784320*n^5 + 13962848*n^4 + 7874944*n^3 + 29641200*n^2 - 2080800*n + 2027025.
Row 10: a(n) = 9728*n^9 + 102144*n^8 + 2830848*n^7 + 9645312*n^6 + 98382912*n^5 + 106720416*n^4 + 522283552*n^3 + 69265488*n^2 + 255468870*n - 34459425.

A167572 is the ED3 array.
A000012, A016969, A167573, A167574 and A167575 equal the first five rows of the ED3 array.
A014480, A167581, A167582, A168305 and A168306 equal the first five left hand triangle columns.
A001147 equals the first right hand triangle column.
A167576 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Comment and links added by Johannes W. Meijer, Nov 23 2009

A117303 Self-inverse permutation of the natural numbers based on the bijection (2x-1)2^(y-1) <--> (2y-1)2^(x-1).

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 7, 16, 12, 32, 10, 64, 24, 128, 9, 256, 48, 512, 20, 1024, 96, 2048, 14, 4096, 192, 8192, 40, 16384, 384, 32768, 11, 65536, 768, 131072, 80, 262144, 1536, 524288, 28, 1048576, 3072, 2097152, 160, 4194304, 6144, 8388608, 18, 16777216

Offset: 1

Reinhard Zumkeller, Apr 24 2006

nonn

a(a(n)) = n; fixed points A014480: a(A014480(n)) = A014480(n). - Reinhard Zumkeller, Apr 27 2006

Cf. A000265, A007814, A006519, A005408, A000079.

Cf. A118413, A118416.

a:= n-> (j-> (2*j+1)*2^((n/2^j-1)/2))(padic[ordp](n, 2)):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 23 2019

a[n_] := (2 IntegerExponent[2 n, 2] - 1)*2^((n/2^IntegerExponent[n, 2] + 1)/2 - 1); Array[a, 50] (* Jean-François Alcover, Mar 12 2019 *)

def A117303(n): return (((m:=(n&-n).bit_length())<<1)-1)*(1<<(n>>m)) # Chai Wah Wu, Jul 14 2022

a(n) = (2*A001511(n) - 1) * 2^(A003602(n) - 1).

Spelling corrected by Jason G. Wurtzel, Aug 23 2010

A126984 Expansion of 1/(1+2xc(x)), c(x) the g.f. of Catalan numbers A000108.

Original entry on oeis.org

1, -2, 2, -4, 2, -12, -12, -72, -190, -700, -2308, -8120, -28364, -100856, -360792, -1301904, -4727358, -17268636, -63405012, -233885784, -866327748, -3220976616, -12016209192, -44966763504, -168750724428, -634935132312, -2394717424552, -9051945482032

Offset: 0

Philippe Deléham, Mar 21 2007

Hankel transform is (-2)^n.

Hankel transform omitting first term is (-2)^n omitting first term. Hankel transform omitting first two terms is 2*(-1)^n*A014480(n). - Michael Somos, May 16 2022

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

Cf. A000108, A014480, A039599, A268600, A268601.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(2-Sqrt(1-4*x)) )); // G. C. Greubel, May 28 2019

c:=(1-sqrt(1-4*x))/2/x: ser:=series(1/(1+2*x*c),x=0,32): seq(coeff(ser,x,n),n=0..30); # Emeric Deutsch, Mar 24 2007

CoefficientList[Series[1/(2-Sqrt[1-4*x]), {x,0,30}], x] (* G. C. Greubel, May 28 2019 *)

my(x='x+O('x^30)); Vec(1/(2-sqrt(1-4*x))) \\ G. C. Greubel, May 28 2019

(1/(2-sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 28 2019

a(n) = Sum_{k=0..n} A039599(n,k)*(-3)^k.
G.f.: 1/(2 - sqrt(1-4*x)). - G. C. Greubel, May 28 2019
(-1)^n*a(n) = A268600(n) - A268601(n). - Michael Somos, May 16 2022
D-finite with recurrence 3*n*a(n) +2*(-4*n+9)*a(n-1) +8*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024
a(n) = Sum_{k = 0..n} A009766(n-1, k)*(-2)^(n-k) for n >= 1. - Peter Bala, Jun 18 2025

Corrected and extended by Emeric Deutsch, Mar 24 2007

A098503 Triangle T(n,k) by rows: coefficient [x^(n-k)] of 2^n * n! *L(n,1/2,x), with L the generalized Laguerre polynomials in the Abramowitz-Stegun normalization.

Original entry on oeis.org

1, -2, 3, 4, -20, 15, -8, 84, -210, 105, 16, -288, 1512, -2520, 945, -32, 880, -7920, 27720, -34650, 10395, 64, -2496, 34320, -205920, 540540, -540540, 135135, -128, 6720, -131040, 1201200, -5405400, 11351340, -9459450, 2027025, 256, -17408

Offset: 0

Ralf Stephan, Sep 15 2004

			2^0 *0! *L(0,1/2,x) = 1.
2^1 *1! *L(1,1/2,x) = -2*x + 3.
2^2 *2! *L(2,1/2,x) = 4*x^2 - 20*x + 15.
2^3 *3! *L(3,1/2,x) = -8*x^3 + 84*x^2 - 210*x + 105.
2^4 *4! *L(4,1/2,x) = 16*x^4 - 288*x^3 + 1512*x^2 - 2520*x + 945.
Triangle begins:
    1;
   -2,     3;
    4,   -20,    15;
   -8,    84,  -210,     105;
   16,  -288,  1512,   -2520,    945;
  -32,   880, -7920,   27720, -34650,   10395;
   64, -2496, 34320, -205920, 540540, -540540, 135135;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
R. J. Mathar, Gauss-Laguerre and Gauss-Hermite quadrature on 64, 96 and 128 nodes, viXra:1303.0013, Section 3.

Columns include (-1)^n times A000079, n/2*A014480. Diagonals include A001147, -A000906, 4*A001881.

Cf. A223168..A223172 and A223523..A223532.

Table[Reverse[Table[2^n*(-1)^k*n!/k!*Binomial[n + 1/2, n - k], {k, 0, n}]], {n, 0, 7}] (* T. D. Noe, Apr 05 2013 *)

T(n, k) = (-2)^n * (-1)^k * n!/(n-k)! * binomial(n+1/2,k), = (-1)^(n+k) *2^(n-2k) *k! *binomial(2n+1,2k)*binomial(2k,k), n>=0, k<=n.

A093968 Inverse binomial transform of n*Pell(n).

Original entry on oeis.org

0, 1, 2, 6, 8, 20, 24, 56, 64, 144, 160, 352, 384, 832, 896, 1920, 2048, 4352, 4608, 9728, 10240, 21504, 22528, 47104, 49152, 102400, 106496, 221184, 229376, 475136, 491520, 1015808, 1048576, 2162688, 2228224, 4587520, 4718592, 9699328, 9961472, 20447232, 20971520

Offset: 0

Paul Barry, Apr 21 2004

Binomial transform is A093967.
Binomial transform of (-1)^(n+1)(n*Pell(n-2)) (see A093969).
S-D transform of A001477 (cf. A051159). - Philippe Deléham, Aug 01 2006
a(n) is also the number of projective permutations of vertices of regular n-gons. A permutation of n vertices (AFB...CD) is considered 'projective' if there exists a line so that all the vertices can be projected onto it and the resulted points can be read in the same order: A'F'B'...C'D'. - Anton Zakharov, Jul 25 2016

			a(3) = 6, as there are only 6 projective permutations of vertices in a triangle ABC: ABC,CBA,ACB,BCA,CAB,BAC and it is equal to the number of simple permutations of three elements.
a(4) = 8, as there are only 8 permutations of vertices in a square, satisfying the projective criterion: ADBC,DACB,DCAB,CDBA,CBDA,BCAD,BACD,ABDC. ADCB is not allowed, cause there is no way to draw a line so that the projections A'B'C'D' of the original points form a line segment B'C' lying inside A'D' on this line. - _Anton Zakharov_, Jul 25 2016

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

Cf. A000129, A132314.

a[n_] := n*2^Floor[(n - 1)/2]; Array[a, 40, 0] (* Amiram Eldar, Feb 13 2023 *)

G.f.: x(1+2x+2x^2)/(1-2x^2)^2;
a(n) = 2^((n-4)/2)n((1+sqrt(2)) + (1-sqrt(2))(-1)^n).
a(2n) = A036289(n). a(2n+1) = A014480(n). - R. J. Mathar, Jun 02 2011
G.f.: x*G(0)/(1-x) where G(k) = 1 + x/(k+1 - 2*x*(k+1)*(k+2)/(2*x*(k+2) + 1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 01 2013
a(n) = n*2^floor((n-1)/2). - Anton Zakharov, Jul 25 2016
E.g.f.: x*(sqrt(2)*sinh(sqrt(2)*x) + 2*cosh(sqrt(2)*x))/2. - Ilya Gutkovskiy, Jul 25 2016
Sum_{n>=1} 1/a(n) = log(2) + sqrt(2)*log(1+sqrt(2)). - Amiram Eldar, Feb 13 2023

A127775 Triangle read by rows: row n consists of n-1 zeros followed by 2n-1.

Original entry on oeis.org

1, 0, 3, 0, 0, 5, 0, 0, 0, 7, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 28 2007

a(A000217(n)) = A005408(n-1), T(n,n) = 2*n - 1. - Reinhard Zumkeller, Feb 11 2007

			First few rows of the triangle are:
1;
0, 3;
0, 0, 5;
0, 0, 0, 7;
...

Cf. A000217, A127648, A127773, A127774, A005408, A014480, A001787.

T(n,k) = (2*n - 1) * 0^(n - k), 1<=k<=n. - Reinhard Zumkeller, Feb 11 2007

More terms from Reinhard Zumkeller, Feb 11 2007

A335919 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), n>=0, max(0,floor(log_2(n))+1)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 8, 6, 20, 16, 4, 40, 56, 32, 1, 68, 152, 144, 64, 94, 376, 480, 352, 128, 114, 844, 1440, 1376, 832, 256, 116, 1744, 4056, 4736, 3712, 1920, 512, 94, 3340, 10856, 15248, 14272, 9600, 4352, 1024, 60, 5976, 27672, 47104, 50784, 40576, 24064

Offset: 0

Alois P. Heinz, Jun 29 2020

Empty external nodes are counted in determining the height of a search tree.

T(n,k) is defined for n,k >= 0. The triangle contains only the positive terms. Terms not shown are zero.

			Triangle T(n,k) begins:
  1;
     1;
        2;
        1, 4;
           6,   8;
           6,  20,   16;
           4,  40,   56,   32;
           1,  68,  152,  144,   64;
               94,  376,  480,  352,  128;
              114,  844, 1440, 1376,  832,  256;
              116, 1744, 4056, 4736, 3712, 1920, 512;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Row sums give A000108.
Column sums give A001699.
Main diagonal gives A011782.
T(n+3,n+2) gives A014480.
T(n,max(0,A000523(n)+1)) = A328349(n).
Cf. A073345, A073429 (another version with 0's), A076615, A195581, A244108, A335920 (the same read by columns), A335921, A335922.

g:= n-> `if`(n=0, 0, ilog2(n)+1):
b:= proc(n, h) option remember; `if`(n=0, 1, `if`(n<2^h,
      add(b(j-1, h-1)*b(n-j, h-1), j=1..n), 0))
    end:
T:= (n, k)-> b(n, k)-`if`(k>0, b(n, k-1), 0):
seq(seq(T(n, k), k=g(n)..n), n=0..12);

g[n_] := If[n == 0, 0, Floor@Log[2, n]+1];
b[n_, h_] := b[n, h] = If[n == 0, 1, If[n < 2^h,
     Sum[b[j - 1, h - 1]*b[n - j, h - 1], {j, 1, n}], 0]];
T[n_, k_] := b[n, k] - If[k > 0, b[n, k - 1], 0];
Table[Table[T[n, k], {k, g[n], n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 08 2021, after Alois P. Heinz *)

Sum_{k=0..n} k * T(n,k) = A335921(n).

Sum_{n=k..2^k-1} n * T(n,k) = A335922(k).

A335920 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), k>=0, k<=n<=2^k-1, read by columns.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 6, 4, 1, 8, 20, 40, 68, 94, 114, 116, 94, 60, 28, 8, 1, 16, 56, 152, 376, 844, 1744, 3340, 5976, 10040, 15856, 23460, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788, 10948, 5096, 1932, 568, 120, 16, 1, 32, 144, 480, 1440, 4056

Offset: 0

Alois P. Heinz, Jun 29 2020

Empty external nodes are counted in determining the height of a search tree.

T(n,k) is defined for n,k >= 0. The triangle contains only the positive terms. Terms not shown are zero.

			Triangle T(n,k) begins:
  1;
     1;
        2;
        1, 4;
           6,   8;
           6,  20,   16;
           4,  40,   56,   32;
           1,  68,  152,  144,   64;
               94,  376,  480,  352,  128;
              114,  844, 1440, 1376,  832,  256;
              116, 1744, 4056, 4736, 3712, 1920, 512;
  ...

Alois P. Heinz, Columns k = 0..10, flattened
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Row sums give A000108.
Column sums give A001699.
Main diagonal gives A011782.
T(n+3,n+2) gives A014480.
T(n,max(0,A000523(n)+1)) = A328349(n).
Cf. A073345, A076615, A195581, A244108, A335919 (the same read by rows), A335921, A335922.

b:= proc(n, h) option remember; `if`(n=0, 1, `if`(n<2^h,
      add(b(j-1, h-1)*b(n-j, h-1), j=1..n), 0))
    end:
T:= (n, k)-> b(n, k)-`if`(k>0, b(n, k-1), 0):
seq(seq(T(n, k), n=k..2^k-1), k=0..6);

b[n_, h_] := b[n, h] = If[n == 0, 1, If[n < 2^h,
     Sum[b[j - 1, h - 1]*b[n - j, h - 1], {j, 1, n}], 0]];
T[n_, k_] := b[n, k] - If[k > 0, b[n, k - 1], 0];
Table[Table[T[n, k], {n, k, 2^k - 1}], {k, 0, 6}] // Flatten (* Jean-François Alcover, Feb 08 2021, after Alois P. Heinz *)

Sum_{k=0..n} k * T(n,k) = A335921(n).

Sum_{n=k..2^k-1} n * T(n,k) = A335922(k).

A113861 a(n) = (1/9)((6n - 7)*2^(n-1) - (-1)^n).

Original entry on oeis.org

0, 1, 5, 15, 41, 103, 249, 583, 1337, 3015, 6713, 14791, 32313, 70087, 151097, 324039, 691769, 1470919, 3116601, 6582727, 13864505, 29127111, 61050425, 127693255, 266571321, 555512263, 1155763769, 2401006023, 4980969017, 10319851975, 21355531833, 44142719431

Offset: 1

N. J. A. Sloane, Jan 25 2006

This sequence is connected with the Collatz problem (see the sequences A045883 and A001045). - Michel Lagneau, Jan 13 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
T. Etzion, On the stopping redundancy of Reed-Muller codes, IEEE Trans. Information Theory 52 (11) (2006) 4867-4879; arXiv:cs.IT/0511056.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A102301.

Cf. A059260, A016813.

Join[{0},Numerator[CoefficientList[Series[(x+1)/((x-1)*(x^2+x-2)),{x,0,40}],x]]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)
LinearRecurrence[{3,0,-4},{0,1,5},40] (* Harvey P. Dale, Jun 16 2022 *)

a(n)=((6*n-7)<<(n-1)-(-1)^n)/9 \\ Charles R Greathouse IV, Jan 13 2012

a(n+1) - 2*a(n) = A001045(n+2), Jacobsthal numbers. - Paul Curtz, Jul 05 2008
3*a(n) - a(n+1) = -1, -2, 4*a(n). - Paul Curtz, Jul 05 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: x^2*(1+2*x)/((1+x)*(1-2*x)^2).
a(n) + a(n+1) = A014480(n-1). (End)
a(n) = 4*a(n-1) - 4*a(n-2) + (-1)^(n+1), n>2. - Gary Detlefs, Dec 19 2010
a(n) = 3*a(n-1) - 4*a(n-3), n>3. - Gary Detlefs, Dec 19 2010
a(n) = n*2^n - A045883(n). - Michel Lagneau, Jan 13 2012
Starting with "1" = triangle A059260 * A016813 as a vector, where A016813 = (4n + 1): [ 1, 5, 9, 13, ...]. - Gary W. Adamson, Mar 06 2012

A118417 a(n) = (2n + 1) 2^(n + 1).

Original entry on oeis.org

2, 12, 40, 112, 288, 704, 1664, 3840, 8704, 19456, 43008, 94208, 204800, 442368, 950272, 2031616, 4325376, 9175040, 19398656, 40894464, 85983232, 180355072, 377487360, 788529152, 1644167168, 3422552064, 7113539584, 14763950080, 30601641984, 63350767616

Offset: 0

Reinhard Zumkeller, Apr 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A014480, A118416.

Cf. A002193, A196525, A195695.

[(2*n+1)*2^(n+1): n in [0..40]]; // Vincenzo Librandi, Dec 26 2010

CoefficientList[Series[2 (1 - 3 x^2 + 2 x^3)/((1 - x)^2 (1 - 2 x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 02 2016 *)
Table[(2n+1)2^(n+1),{n,0,30}] (* or *) LinearRecurrence[{4,-4},{2,12},30] (* Harvey P. Dale, Oct 25 2021 *)

a(n) = A118416(n+1,n) = 2*A014480(n).
G.f.: 2*(1-3*x^2+2*x^3)/((1-x)^2*(1-2*x)^2). - Vincenzo Librandi, Sep 02 2016
Sum_{n>=0} 1/a(n) = A196525. - Fred Daniel Kline, May 24 2019
Sum_{n>=0} (-1)^n/a(n) = arctan(1/sqrt(2))/sqrt(2) = A195695 / A002193. - Amiram Eldar, Oct 01 2022

cp's OEIS Frontend

A167580 A triangle related to the a(n) formulas of the rows of the ED3 array A167572.

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A117303 Self-inverse permutation of the natural numbers based on the bijection (2*x-1)*2^(y-1) <--> (2*y-1)*2^(x-1).

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A126984 Expansion of 1/(1+2*x*c(x)), c(x) the g.f. of Catalan numbers A000108.

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A098503 Triangle T(n,k) by rows: coefficient [x^(n-k)] of 2^n * n! *L(n,1/2,x), with L the generalized Laguerre polynomials in the Abramowitz-Stegun normalization.

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A093968 Inverse binomial transform of n*Pell(n).

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A127775 Triangle read by rows: row n consists of n-1 zeros followed by 2n-1.

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A335919 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), n>=0, max(0,floor(log_2(n))+1)<=k<=n, read by rows.

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A335920 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), k>=0, k<=n<=2^k-1, read by columns.

A117303 Self-inverse permutation of the natural numbers based on the bijection (2x-1)2^(y-1) <--> (2y-1)2^(x-1).

A126984 Expansion of 1/(1+2xc(x)), c(x) the g.f. of Catalan numbers A000108.

A113861 a(n) = (1/9)((6n - 7)*2^(n-1) - (-1)^n).

A118417 a(n) = (2n + 1) 2^(n + 1).