A096053 -id:A096053 - cp's OEIS frontend

A104033 Triangle, read by rows, equal to the matrix inverse of triangle A103327, where A103327(n,k) = binomial(2n+1,2k+1).

1, -3, 1, 25, -10, 1, -427, 175, -21, 1, 12465, -5124, 630, -36, 1, -555731, 228525, -28182, 1650, -55, 1, 35135945, -14449006, 1782495, -104676, 3575, -78, 1, -2990414715, 1229758075, -151714563, 8912475, -305305, 6825, -105, 1, 329655706465, -135565467080, 16724709820, -982532408

Offset: 0

Paul D. Hanna, Feb 28 2005

Column 0 equals signed A009843 (expansion of x/cosh(x)). Row sums form signed A000182 (expansion of tanh(x)).
The matrix logarithm is L(n,k) = -(-1)^(n-k)*A000182(n-k)*A103327(n,k), where A000182 = tangent numbers.
Let E(y) = cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + .... so that 1/E(y) = 1 - 3*y/3! + 25*y^2/5! - 427*y^3/7! + .... Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence (2*n+1)! as defined in Wang and Wang. - Peter Bala, Aug 06 2013

			Rows begin:
1;
-3, 1;
25, -10, 1;
-427, 175, -21, 1;
12465, -5124, 630, -36, 1;
-555731 ,228525, -28182, 1650, -55, 1;
35135945, -14449006, 1782495, -104676, 3575, -78, 1;
-2990414715, 1229758075, -151714563, 8912475, -305305, 6825, -105, 1;
329655706465, -135565467080, 16724709820, -982532408, 33669350, -754936, 11900, -136, 1; ...
From _Peter Bala_, Aug 06 2013: (Start)
The real zeros of the row polynomials R(n,x) seem to converge to the even squares as n increases.
Polynomial |        Real zeros to 6 decimal places
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
R(5,x)     | 3.999986
R(10,x)    | 4.000000, 15.999978
R(15,x)    | 4.000000, 16.000000, 35.999992, 64.414273, 76.998346
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
(End)

W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A000364, A103327, A009843, A000182 (unsigned row sums), A055133, A086645, A086646, A096053, A103364.

{T(n,k) = if(n=j, binomial(2*m-1,2*j-1))))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n, print1(T(n,k),", "));print(""))

{T(n,k) = binomial(2*n+1,2*k+1) * polcoeff(1/cosh(x+x*O(x^(2*n))), 2*n-2*k) * (2*n-2*k)!}
for(n=0,10,for(k=0,n, print1(T(n,k),", "));print(""))

Column k: Sum_{j=0..n} C(2*n+1, 2*j+1) * T(j, k) = 0 (n>k), or 1 (n=k).

Row n: Sum_{j=0..n} T(n, j) * C(2*j+1, 2*k+1) = 0 (k

Sum_{k=0..n} T(n, k) * 4^k = 1 for n >= 0.

T(n, k) = (-1)^(n-k)*A000364(n-k)*A103327(n, k), where A000364 = Euler numbers.

Sum_{k=0..n} (-1)^(n-k)*T(n, k) = A002084(n). - Philippe Deléham, Aug 27 2005

From Peter Bala, Aug 06 2013: (Start)

Generating function: 1/sqrt(x)*sinh(sqrt(x)*t)/cosh(t) = t + (-3 + x)*t^3/3! + (25 - 10*x + x^2)*t^5/5! + ....

Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} binomial(2*n+1,2*k+1)*R(k,x) with initial value R(0,x) = 1.

It appears that for arbitrary nonzero complex x we have

lim_{n -> inf} R(n,x^2)/R(n,0) = (1/(Pi/2*x))*sin(Pi/2*x).

A stronger result than pointwise convergence may hold: the convergence may be uniform on compact subsets of the complex plane. This would explain the observation that the real zeros of the polynomials R(n,x) seem to converge to the even squares 4, 16, 36, ... as n increases. Some numerical examples are given below. Cf. A055133, A086646 and A103364.

If p = 2*n + 1 is a prime then all the entries in row n are divisible by p, apart from T(n,n) = 1. Thus the row sum is congruent to 1 modulo p.

Row sums R(n,1) = (-1)^n*A000182(n+1).

R(n,4) = 1; R(n,16) = (1/2)*( 3^(2*n+1) - 1 ) = A096053(n);

R(n,36) = (1/3)*( 5^(2*n+1) - 3^(2*n+1) + 1 );

R(n,64) = (1/4)*( 7^(2*n+1) - 5^(2*n+1) + 3^(2*n+1) - 1 ). (End)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

A107903 Generalized NSW numbers.

Original entry on oeis.org

1, 10, 76, 568, 4240, 31648, 236224, 1763200, 13160704, 98232832, 733219840, 5472827392, 40849739776, 304906608640, 2275853910016, 16987204845568, 126794223124480, 946404965613568, 7064062832410624, 52726882796830720

Offset: 0

Paul Barry, May 27 2005

Counts total area under elevated Schroeder paths of length 2n+2, where horizontal steps can choose from three colors.
Case r=3 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315 and case r=4 gives NSW numbers A096053.
Fifth binomial transform of (1+8x)/(1-16x^2), A107906.
If p is an odd prime, a((p-1)/2) == 1 mod p. - Altug Alkan, Mar 17 2016

Index entries for linear recurrences with constant coefficients, signature (8,-4).

Cf. A001834, A099156, A102591.

Table[Sum[Binomial[2 n + 1, 2 k] 3^k, {k, 0, n}], {n, 0, 20}] (* or *) CoefficientList[Series[(1 + 2 x)/(1 - 8 x + 4 x^2), {x, 0, 20}], x] (* Michael De Vlieger, Mar 17 2016 *)

Vec((1+2*x)/(1-8*x+4*x^2) + O(x^40)) \\ Michel Marcus, Mar 17 2016

G.f.: (1+2*x)/(1-8*x+4*x^2). [corrected by Ralf Stephan, Nov 30 2010]
a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k)*3^k.
a(n) = ((1+sqrt(3))*(4+2*sqrt(3))^n+(1-sqrt(3))*(4-2*sqrt(3))^n)/2 = A099156(n+1)+2*A099156(n).
a(n) = 8*a(n-1) - 4*a(n-2); a(0) = 1, a(1) = 10. - Lekraj Beedassy, Apr 19 2020
a(n) = 2^n*A001834(n). - Philippe Deléham, Mar 18 2023

Typo corrected and link added by Johannes W. Meijer, Aug 07 2010

A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.

Original entry on oeis.org

1, 91, 3751, 138811, 5028751, 181308931, 6529545751, 235085301451, 8463265086751, 304679288612371, 10968470088963751, 394865064451017691, 14215143591303768751, 511745180725868773411, 18422826609078989373751, 663221758853362301815531, 23875983327059668074930751

Offset: 1

Benoit Cloitre, Jun 18 2004

Colin Barker, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (50,-553,1800,-1296).

Cf. A083420, A096045, A096046, A096047, A096048, A096053.

a[n_] := 6^(2*n-1) * BernoulliB[2*n, 1/6] / BernoulliB[2*n]; Array[a, 15] (* Amiram Eldar, May 07 2025 *)

a(n)=(1/12)*36^n-(1/6)*9^n-(1/4)*4^n+1/2;

a(n) = (1/12)*(36^n - 2*9^n - 3*4^n+6).
From Colin Barker, May 30 2020: (Start)
G.f.: x*(1 - 6*x)*(1 + 47*x + 36*x^2) / ((1 - x)*(1 - 4*x)*(1 - 9*x)*(1 - 36*x)).
a(n) = 50*a(n-1) - 553*a(n-2) + 1800*a(n-3) - 1296*a(n-4) for n>4. (End)

A164907 a(n) = (3*3^n-(-1)^n)/2.

Original entry on oeis.org

1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165

Offset: 0

Klaus Brockhaus, Aug 31 2009

Interleaving of A096053 and A083884 without initial term 1.
Partial sums are (essentially) in A080926.
First differences are (essentially) in A105723.
a(n)+a(n+1) = A008776(n+1) = A099856(n+1) = A110593(n+2).
Binomial transform of A056450. Inverse binomial transform of A164908.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,3).

Equals A046717 without initial term 1 and A080925 without initial term 0. Equals A084182 / 2 from second term onward.

Cf. A096053, A083884, A080926, A105723, A008776, A099856, A110593, A056450, A164908.

Magma
```
[ (3*3^n-(-1)^n)/2: n in [0..25] ];
```

A164907:=n->(3*3^n - (-1)^n)/2; seq(A164907(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

Table[(3*3^n - (-1)^n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Mar 21 2014 *)
LinearRecurrence[{2,3},{1,5},50] (* Harvey P. Dale, Oct 31 2018 *)

a(n) = 2*a(n-1)+3*a(n-2) for n > 1; a(0) = 1, a(1) = 5.
G.f.: (1+3*x)/((1+x)*(1-3*x)).
a(n) = 3*a(n-1)+2*(-1)^n. - Carmine Suriano, Mar 21 2014

A107906 Expansion of (1+8x)/(1-16x^2).

Original entry on oeis.org

1, 8, 16, 128, 256, 2048, 4096, 32768, 65536, 524288, 1048576, 8388608, 16777216, 134217728, 268435456, 2147483648, 4294967296, 34359738368, 68719476736, 549755813888, 1099511627776, 8796093022208, 17592186044416

Offset: 0

Paul Barry, May 27 2005

Fifth binomial transform is A096053.

Index entries for linear recurrences with constant coefficients, signature (0,16).

a(n) = ((1+sqrt(4))*(2*sqrt(4))^n + (1-sqrt(4))*(-2*sqrt(4))^n)/2;
a(n) = 3*4^n/2 - (-4)^n/2.
a(2n) = 16^n, a(2n+1) = 8*16^n.

A255043 a(n) = (5*9^n - 1)/2.

Original entry on oeis.org

2, 22, 202, 1822, 16402, 147622, 1328602, 11957422, 107616802, 968551222, 8716961002, 78452649022, 706073841202, 6354664570822, 57191981137402, 514727830236622, 4632550472129602, 41692954249166422, 375236588242497802, 3377129294182480222

Offset: 0

L. Edson Jeffery, Feb 13 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A096053, A138894, A191681, A198964, A198969.

Row 2 of A255044.

[(5*9^n -1)/2: n in [0..20]]; // G. C. Greubel, Feb 07 2021

Table[(5*9^n - 1)/2, {n, 0, 19}]
LinearRecurrence[{10,-9},{2,22},20] (* Harvey P. Dale, Jun 15 2018 *)

[(5*9^n -1)/2 for n in (0..20)] # G. C. Greubel, Feb 07 2021

G.f.: 2*(1+x)/((1-x)*(1-9*x)).
Recurrence: a(n) = 10*a(n-1) - 9*a(n-2), n>=2, a(0) = 2, a(1) = 22.
a(n) = 2*A138894(n).
E.g.f.: (5*exp(9*x) - exp(x))/2. - G. C. Greubel, Feb 07 2021

A320030 Automaton sum similar to A102376 but using mod 3 instead of mod 2.

Original entry on oeis.org

1, 4, 13, 4, 16, 52, 13, 52, 121, 4, 16, 52, 16, 64, 208, 52, 208, 484, 13, 52, 121, 52, 208, 484, 121, 484, 1093, 4, 16, 52, 16, 64, 208, 52, 208, 484, 16, 64, 208, 64, 256, 832, 208, 832, 1936, 52, 208, 484, 208, 832, 1936, 484, 1936, 4372, 13, 52, 121, 52

Offset: 1

Nathan M Epstein, Dec 10 2018

nonn

The automaton that generates this sequence operates on a grid of cells c(i,j). The cells have three possible values, 0, 1, and 2. The next generation in the CA is calculated by applying the following rule to each cell: c(i,j) = ( c(i+1,j-1) + c(i+1,j+1) + c(i-1,j-1) + c(i-1,j+1) ) mod 3.
Start with a single cell with a value of 1, with all other cells set to 0. For each generation, the term in this sequence c(n) is the aggregate values of all cells in the grid for each discrete generation of the automaton (i.e., not cumulative over multiple generations).
The cellular automaton that generates this sequence has been empirically observed to repeat the number of active cells (4 in this case) if the iteration number N is a power of the modulus + 1. The modulus in this case is 3.
This has been observed to occur with any prime modulus and any starting pattern of cells. I'm picking this particular implementation because it's the same as the one used in A102376.
Counting the active (nonzero) cells instead of taking the sum also creates a different but related sequence. This sequence is the sum of each iteration, and cells in this automaton have values 0, 1, or 2. Only for mod 2 are both the sum and active cell counts the same.

Nathan M Epstein, Animation of CA

Cf. A096053.

Cf. A102376 (mod 2), A320100 (mod 5).

import numpy as np
from scipy import signal
frameSize = 301
filter = [[0,1,0],[1,0,1],[0,1,0]] # this defines the CA neighborhood
frame  = np.zeros((frameSize,frameSize))
frame[frameSize//2,frameSize//2] = 1
mod = 3
sequence = [1]
for j in range(140):
    frame = signal.convolve2d(frame, filter, mode='same')
    frame = np.mod(frame, mod)
    sequence.append(int(np.sum(frame.reshape(1,-1))))

a(3^n) = A096053(n).

A198960 a(n) = 3*9^n-1.

Original entry on oeis.org

2, 26, 242, 2186, 19682, 177146, 1594322, 14348906, 129140162, 1162261466, 10460353202, 94143178826, 847288609442, 7625597484986, 68630377364882, 617673396283946, 5559060566555522, 50031545098999706, 450283905890997362

Offset: 0

Vincenzo Librandi, Nov 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Magma
```
[3*9^n-1: n in [0..20]];
```

CoefficientList[Series[2*(1 + 3*x)/(1 - 10*x + 9*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 03 2013 *)
LinearRecurrence[{10,-9},{2,26},30] (* Harvey P. Dale, Dec 07 2024 *)

a(n) = 2*A096053(n).
a(n) = 9*a(n-1)+8, n>0.
a(n) = 10*a(n-1)-9*a(n-2), n>1.
G.f.: 2*(1 + 3*x)/(1 - 10*x + 9*x^2). - Vincenzo Librandi, Jan 03 2013

Showing 1-10 of 11 results. Next

cp's OEIS Frontend

A138894 Expansion of (1+x)/(1-10*x+9*x^2).

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A191681 a(n) = (9^n - 1)/2.

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A104033 Triangle, read by rows, equal to the matrix inverse of triangle A103327, where A103327(n,k) = binomial(2*n+1,2*k+1).

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A107903 Generalized NSW numbers.

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A096054 a(n) = (36^n/6)*B(2n,1/6)/B(2n) where B(n,x) is the n-th Bernoulli polynomial and B(k) = B(k,0) is the k-th Bernoulli number.

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A164907 a(n) = (3*3^n-(-1)^n)/2.

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A107906 Expansion of (1+8x)/(1-16x^2).

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A138894 Expansion of (1+x)/(1-10x+9x^2).

A104033 Triangle, read by rows, equal to the matrix inverse of triangle A103327, where A103327(n,k) = binomial(2n+1,2k+1).