A059826 -id:A059826 - cp's OEIS frontend

A001699 Number of binary trees of height n; or products (ways to insert parentheses) of height n when multiplication is non-commutative and non-associative.

1, 1, 3, 21, 651, 457653, 210065930571, 44127887745696109598901, 1947270476915296449559659317606103024276803403, 3791862310265926082868235028027893277370233150300118107846437701158064808916492244872560821

Offset: 0

N. J. A. Sloane and Jeffrey Shallit

Approaches 1.5028368...^(2^n), see A077496. Row sums of A065329 as square array. - Henry Bottomley, Oct 29 2001. Also row sum of square array A073345 (AK).

			G.f. = 1 + x + 3*x^2 + 21*x^3 + 651*x^4 + 457653*x^5 + ... - _Michael Somos_, Jun 02 2019

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 307.
I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.
I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. Part II is by A. Erdelyi and I. M. H. Etherington, and is on pages vii-xiv of the same issue.
T. K. Moon, Enumerations of binary trees, types of trees and the number of reversible variable length codes, submitted to Discrete Applied Mathematics, 2000.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

David Wasserman, Table of n, a(n) for n = 0..12 [Shortened file because terms grow rapidly: see Wasserman link below for an additional term]
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Mayfawny Bergmann, Efficiency of Lossless Compression of a Binary Tree via its Minimal Directed Acyclic Graph Representation. Rose-Hulman Undergraduate Mathematics Journal: Vol. 15 : Iss. 2, Article 1. (2014).
Henry Bottomley, Illustration of initial terms
I. M. H. Etherington, Non-associate powers and a functional equation, Math. Gaz. 21 (1937), 36-39; addendum 21 (1937), 153.
I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162. [Annotated scanned copy]
Samuele Giraudo, The combinator M and the Mockingbird lattice, arXiv:2204.03586 [math.CO], 2022.
Claude Lenormand, Arbres et permutations II, see p. 6
David E. Narváez, Proof of polynomial recursive equation of order 2, Jun 05 2025.
David Wasserman, Table of n, a(n) for n = 0..13
Eric Weisstein's World of Mathematics, Binary Tree
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for sequences related to parenthesizing
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Row sums of A065329.
Column sums of A335919, A335920.
Cf. A002658, A056207, A002449, A003095.
Cf. A004019, A077496, A076949, A213437.

s := proc(n) local i,j,ans; ans := [ 1 ]; for i to n do ans := [ op(ans),2*(add(j,j=ans)-ans[ i ])*ans[ i ]+ans[ i ]^2 ] od; RETURN(ans); end; s(10);

a[0] = 1; a[n_] := a[n] = 2*a[n-1]*Sum[a[k], {k, 0, n-2}] + a[n-1]^2; Table[a[n], {n, 0, 9}] (* Jean-François Alcover, May 16 2012 *)
a[ n_] := If[ n < 2, Boole[n >= 0], With[{u = a[n - 1], v = a[n - 2]}, u (u + v + u/v)]]; (* Michael Somos, Jun 02 2019 *)

{a(n) = if( n<=1, n>=0, a(n-1) * (a(n-1) + a(n-2) + a(n-1) / a(n-2)))}; /* Michael Somos, 2000 */

from functools import lru_cache
@lru_cache(maxsize=None)
def a(n): return 1 if n <= 1 else a(n-1) * (a(n-1) + a(n-2) + a(n-1)//a(n-2))
print([a(n) for n in range(10)]) # Michael S. Branicky, Nov 10 2022 after Michael Somos

a(n+1) = 2*a(n)*(a(0) + ... + a(n-1)) + a(n)^2.
a(n+1) = a(n)^2 + a(n) + a(n)*sqrt(4*a(n)-3), if n > 0.
a(n) = A003095(n+1) - A003095(n) = A003095(n)^2 - A003095(n) + 1. - Henry Bottomley, Apr 26 2001; offset of LHS corrected by Anindya Bhattacharyya, Jun 21 2013
a(n) = A059826(A003095(n-1)).
From Peter Bala, Feb 03 2017: (Start)
a(n) = Product_{k = 1..n} A213437(k).
a(n) + a(n-1) = A213437(n+1) - A213437(n). (End)
a(n) = -a(n-2)^3 + a(n-1)^2 + 3*a(n-1)*a(n-2) + 2*a(n-2)^2 + 2*a(n-1) - 4*a(n-2) (see Narváez link for proof). - Boštjan Gec, Oct 10 2024

Minor edits by Vaclav Kotesovec, Oct 04 2014

A082044 Main diagonal of A082043: a(n) = n^4 + 2*n^2 + 1.

Original entry on oeis.org

1, 4, 25, 100, 289, 676, 1369, 2500, 4225, 6724, 10201, 14884, 21025, 28900, 38809, 51076, 66049, 84100, 105625, 131044, 160801, 195364, 235225, 280900, 332929, 391876, 458329, 532900, 616225, 708964, 811801, 925444, 1050625, 1188100

Offset: 0

Paul Barry, Apr 03 2003

a(n) = longest side b of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (n^2+2, n^4+2*n^2+1, n^4+3*n^2+1).

			G.f. = 1 + 4*x + 25*x^2 + 100*x^3 + 289*x^4 + 676*x^5 + 1369*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A002061 A002522, A059826, A082043, A082047.

See A120062 for sequences related to integer-sided triangles with integer inradius n.

[(n^2+1)^2: n in [0..40]]; // G. C. Greubel, Dec 24 2022

seq(fibonacci(3,n)^2,n=0..33); # Zerinvary Lajos, Apr 09 2008

Fibonacci[3,Range[0,40]]^2 (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,4,25,100,289},40] (* Harvey P. Dale, Feb 27 2015 *)

a(n) = n^4+2*n^2+1; \\ Michel Marcus, Jan 22 2016

[(n^2+1)^2 for n in range(41)] # G. C. Greubel, Dec 24 2022

a(n) = n^4 + 2*n^2 + 1.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Feb 27 2015
a(n) = (4*A000217(n-1)^2 + 2*A002061(n))^2 / a(n-1). - Bruce J. Nicholson, Apr 17 2017
a(n) = A002522(n)^2 = (n^2 + 1)^2 = a(-n) for all n in Z. - Michael Somos, Apr 17 2017
G.f.: (1 - x + 15*x^2 + 5*x^3 + 4*x^4) / (1 - x)^5. - Michael Somos, Apr 17 2017
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=0} 1/a(n) = Pi^2*csch(Pi)^2/4 + Pi*coth(Pi)/4 + 1/2.
Sum_{n>=0} (-1)^n/a(n) = Pi^2*csch(Pi)*coth(Pi)/4 + Pi*csch(Pi)/4 + 1/2. (End)
E.g.f.: (1 + 3*x + 9*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 24 2022

A100019 a(n) = n^4 + n^3 + n^2.

Original entry on oeis.org

0, 3, 28, 117, 336, 775, 1548, 2793, 4672, 7371, 11100, 16093, 22608, 30927, 41356, 54225, 69888, 88723, 111132, 137541, 168400, 204183, 245388, 292537, 346176, 406875, 475228, 551853, 637392, 732511, 837900, 954273, 1082368, 1222947

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 19 2004

a(n) are the numbers m such that: j^2 = j + m + sqrt(j*m) with corresponding numbers j given by A002061(n+1), and with sqrt(j*m) = A027444(n) = n* A002061(n+1). - Richard R. Forberg, Sep 03 2013.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000583, A002523, A091940, A071253, A059826, A027445, A053699.

[n^4+n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

A100019:=n->n^4+n^3+n^2: seq(A100019(n), n=0..50); # Wesley Ivan Hurt, Apr 14 2017

f[n_]:=n^4+n^3+n^2;Array[f,50,0] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)

a(n) = n^4 + n^3 + n^2 \\ Indranil Ghosh, Apr 15 2017

From Indranil Ghosh, Apr 15 2017: (Start)
G.f.: -x(3 + 13x + 7x^2 + x^3)/(x - 1)^5
E.g.f.: exp(x)*x*(3 + 11x + 7x^2 + x^3)
(End)

A123866 a(n) = n^6 - 1.

Original entry on oeis.org

0, 63, 728, 4095, 15624, 46655, 117648, 262143, 531440, 999999, 1771560, 2985983, 4826808, 7529535, 11390624, 16777215, 24137568, 34012223, 47045880, 63999999, 85766120, 113379903, 148035888, 191102975, 244140624, 308915775, 387420488

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 7 = 0 iff n mod 7 > 0: a(A008589(n))=6; a(A047304(n)) = 0; a(n) mod 7 = 6*(1-A082784(n)).

a(n) = A005563(n-1)*A059826(n) = A068601(n)*A001093(n). - Reinhard Zumkeller, Feb 02 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A024004, A001014, A005563, A123865, A123867, A123868.

List([1..30], n-> n^6 -1); # G. C. Greubel, Aug 08 2019

a123866 = (subtract 1) . (^ 6)  -- Reinhard Zumkeller, Mar 11 2014

[n^6 - 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

A123866:=n->n^6 - 1; seq(A123866(n), n=1..40); # Wesley Ivan Hurt, Feb 26 2014

Table[n^6-1,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0,63,728,4095, 15624,46655, 117648}, 30] (* Harvey P. Dale, Nov 18 2012 *)

A123866(n):=n^6 -1 $ makelist(A123866(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n)=n^6-1 \\ Charles R Greathouse IV, Oct 07 2015

[n^6 -1 for n in (1..30)] # G. C. Greubel, Aug 08 2019

G.f.: x^2*(63 + 287*x + 322*x^2 + 42*x^3 + 7*x^4 - x^5)/(1-x)^7. - Colin Barker, May 08 2012
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(1)=0, a(2)=63, a(3)=728, a(4)=4095, a(5)=15624, a(6)=46655, a(7)=117648. - Harvey P. Dale, Nov 18 2012
Sum_{n>=2} 1/a(n) = 11/12 - Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/6. - Vaclav Kotesovec, Feb 14 2015
E.g.f.: 1 + (-1 + x + 31*x^2 + 90*x^3 + 65*x^4 + 15*x^5 + x^6)*exp(x). - G. C. Greubel, Aug 08 2019
Product_{n>=2} (1 + 1/a(n)) = 6*Pi^2*sech(sqrt(3)*Pi/2)^2. - Amiram Eldar, Jan 20 2021

A082039 Symmetric square array defined by T(n,k) = k^2n^2 + kn + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 13, 21, 13, 1, 1, 21, 43, 43, 21, 1, 1, 31, 73, 91, 73, 31, 1, 1, 43, 111, 157, 157, 111, 43, 1, 1, 57, 157, 241, 273, 241, 157, 57, 1, 1, 73, 211, 343, 421, 421, 343, 211, 73, 1, 1, 91, 273, 463, 601, 651, 601, 463, 273, 91, 1, 1, 111, 343, 601, 813, 931, 931, 813, 601, 343, 111, 1

Offset: 0

Paul Barry, Apr 02 2003

			Square array T(n,k) begins:
  1  1  1   1   1   1 ...
  1  3  7  13  21  31 ...
  1  7 21  43  73 111 ...
  1 13 43  91 157 241 ...
  1 21 73 157 273 421 ...
  ...

Rows include A054569, A002061, A082040, A082041.
Main diagonal is A059826.
Cf. A082038.

A131471 a(n) = n^5+n.

Original entry on oeis.org

0, 2, 34, 246, 1028, 3130, 7782, 16814, 32776, 59058, 100010, 161062, 248844, 371306, 537838, 759390, 1048592, 1419874, 1889586, 2476118, 3200020, 4084122, 5153654, 6436366, 7962648, 9765650, 11881402, 14348934, 17210396, 20511178, 24300030, 28629182, 33554464

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

Cf. A271208, A271209.

[n^5+n: n in [0..30]]; // Vincenzo Librandi, Oct 01 2011

Table[n^5 + n, {n, 0, 50}] (* Paolo Xausa, Nov 03 2024 *)

a(n) = n^5+n; \\ Michel Marcus, Apr 02 2016

G.f.: 2*x*(1+11*x+36*x^2+11*x^3+x^4)/(-1+x)^6. - R. J. Mathar, Nov 14 2007

a(n) = A271208(n) + 1 = A271209(n) - 1. - Paolo Xausa, Nov 03 2024

A219069 Triangle read by rows: T(n,k) = n^4 + (n*k)^2 + k^4, 1 <= k <= n.

Original entry on oeis.org

3, 21, 48, 91, 133, 243, 273, 336, 481, 768, 651, 741, 931, 1281, 1875, 1333, 1456, 1701, 2128, 2821, 3888, 2451, 2613, 2923, 3441, 4251, 5461, 7203, 4161, 4368, 4753, 5376, 6321, 7696, 9633, 12288, 6643, 6901, 7371, 8113, 9211, 10773, 12931, 15841, 19683

Offset: 1

Reinhard Zumkeller, Nov 11 2012

Entry 17a from July 9, 1796 in Gauss's Mathematical Diary: "Summa trium quadratorum continue proportionalium numquam primus esse potest: conspicuum exemplum novimus et quod congruum videtur. Confidamus." Paul Bachmann explains that this note is based on Gauss's discovery of this factorization: n^4 + n^2*k^2 + k^4 = (n^2 + n*k + k^2) * (n^2 - n*k + k^2).

			The triangle begins:
.  1:      3
.  2:     21    48
.  3:     91   133   243
.  4:    273   336   481   768
.  5:    651   741   931  1281  1875
.  6:   1333  1456  1701  2128  2821  3888
.  7:   2451  2613  2923  3441  4251  5461  7203
.  8:   4161  4368  4753  5376  6321  7696  9633 12288
.  9:   6643  6901  7371  8113  9211 10773 12931 15841 19683
. 10:  10101 10416 10981 11856 13125 14896 17301 20496 24661 30000
. 11:  14763 15141 15811 16833 18291 20293 22971 26481 31003 36741 43923

Carl Friedrich Gauss (Hans Wussing, ed.), Mathematisches Tagebuch 1796-1814, Ostwalds Klassiker der Exakten Wissenschaften, Leipzig (1976, 1979), pp. 43, 63, 90.

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Wikipedia, Gauss's diary
Wikipedia, Paul Gustav Heinrich Bachmann

Cf. A059826 (left edge), A219056 (right edge), A219070 (row sums).
Cf. A239426 (central terms).
Cf. A243201 (diagonal (n + 1, n)). - Mathew Englander, Jun 03 2014

a219069 n k = a219069_tabl !! (n-1) !! (k-1)
a219069_row n = a219069_tabl !! n
a219069_tabl = zipWith (zipWith (*)) a215630_tabl a215631_tabl

Table[n^4+(n*k)^2+k^4,{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jul 05 2020 *)

T(n,k) = A215630(n,k) * A215631(n,k), 1 <= k <= n.

A131472 a(n) = n^6 + n.

Original entry on oeis.org

0, 2, 66, 732, 4100, 15630, 46662, 117656, 262152, 531450, 1000010, 1771572, 2985996, 4826822, 7529550, 11390640, 16777232, 24137586, 34012242, 47045900, 64000020, 85766142, 113379926, 148035912, 191103000, 244140650, 308915802

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

[n^6+n: n in [0..30]]; // _Vincenzo Librandi+, Oct 01 2011

Table[n^6+n,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, May 12 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,2,66,732,4100,15630,46662},60] (* Harvey P. Dale, May 03 2012 *)

G.f.: 2*x*(1 + 26*x + 156*x^2 + 146*x^3 + 31*x^4)/(1 - x)^7. - R. J. Mathar, Nov 14 2007
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(0)=0, a(1)=2, a(2)=66, a(3)=732, a(4)=4100, a(5)=15630, a(6)=46662. - Harvey P. Dale, May 03 2012
E.g.f.: exp(x)*x*(2 + 31*x + 90*x^2 + 65*x^3 + 15*x^4 + x^5). - Stefano Spezia, Oct 08 2022

A059848 As a square table by antidiagonals, the n-digit number which in base k starts 1010101...

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 0, 0, 1, 3, 5, 2, 1, 0, 1, 4, 10, 10, 3, 0, 0, 1, 5, 17, 30, 21, 3, 1, 0, 1, 6, 26, 68, 91, 42, 4, 0, 0, 1, 7, 37, 130, 273, 273, 85, 4, 1, 0, 1, 8, 50, 222, 651, 1092, 820, 170, 5, 0, 0, 1, 9, 65, 350, 1333, 3255, 4369, 2460, 341, 5, 1, 0, 1, 10

Offset: 0

Henry Bottomley, Feb 26 2001

			T(5,3)=10101 base 3=81+9+1=91; T(4,6)=1010 base 6=216+6=222. Table starts {0,0,0,0,...}, {1,1,1,1,...}, {0,1,2,3,...}, {1,2,5,10,...}, ...

Rows include A000004, A000012, A001477, A002522, A034262, A059826, A059830, A059839. Columns include A000035, A008619, A000975, A033113, A033114, A033115, A033116, A033117, A033118, A033119, A056830.

T(n, k)=floor[k^(n+1)/(k^2-1)] =T(n-2, k)+k^(n-1) =k*T(n-1, k)-((-1)^n-1)/2

A100606 a(n) = n^4 + n^3 + n.

Original entry on oeis.org

0, 3, 26, 111, 324, 755, 1518, 2751, 4616, 7299, 11010, 15983, 22476, 30771, 41174, 54015, 69648, 88451, 110826, 137199, 168020, 203763, 244926, 292031, 345624, 406275, 474578, 551151, 636636, 731699, 837030, 953343, 1081376, 1221891, 1375674, 1543535, 1726308

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 30 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000583, A002523, A027445, A053699, A059826, A071253, A091940, A100019.

[n^4+n^3+n: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

Table[n^4+n^3+n,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,3,26,111,324},40] (* Harvey P. Dale, Apr 25 2015 *)

a(n)=n^4+n^3+n \\ Charles R Greathouse IV, Oct 21 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=0, a(1)=3, a(2)=26, a(3)=111, a(4)=324. - Harvey P. Dale, Apr 25 2015
From Elmo R. Oliveira, Aug 29 2025: (Start)
G.f.: x*(3 + 11*x + 11*x^2 - x^3)/(1-x)^5.
E.g.f.: x*(3 + 10*x + 7*x^2 + x^3)*exp(x). (End)

cp's OEIS Frontend

A001699 Number of binary trees of height n; or products (ways to insert parentheses) of height n when multiplication is non-commutative and non-associative.

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A082044 Main diagonal of A082043: a(n) = n^4 + 2*n^2 + 1.

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A100019 a(n) = n^4 + n^3 + n^2.

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A123866 a(n) = n^6 - 1.

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A082039 Symmetric square array defined by T(n,k) = k^2*n^2 + k*n + 1, read by antidiagonals.

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A131471 a(n) = n^5+n.

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A219069 Triangle read by rows: T(n,k) = n^4 + (n*k)^2 + k^4, 1 <= k <= n.

A082039 Symmetric square array defined by T(n,k) = k^2n^2 + kn + 1, read by antidiagonals.