A015083 -id:A015083 - cp's OEIS frontend

A015108 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-11.

1, 1, -10, -1231, 1636130, 23957879562, -3858392581773300, -6835385537899011365535, 133202313157282627679850238250, 28553099061411464607955930776882965774

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 10*x^2 - 1231*x^3 + 1636130*x^4 + 23957879562*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..44
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. this sequence (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
Column k=11 of A290789.

m = 10; ContinuedFractionK[If[i == 1, 1, -(-11)^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
  ary
end
def A015108(n)
  A(-11, n)
end # Seiichi Manyama, Dec 25 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-11 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-11*x)) = 1/(1-x/(1+11*x/(1-11^2*x/(1+11^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 28 2016

Offset changed to 0 by Seiichi Manyama, Dec 25 2016

A290759 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - kx/(1 - k^2x/(1 - k^3x/(1 - k^4x/(1 - ...)))))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 17, 14, 1, 1, 1, 5, 43, 171, 42, 1, 1, 1, 6, 89, 1252, 3113, 132, 1, 1, 1, 7, 161, 5885, 104098, 106419, 429, 1, 1, 1, 8, 265, 20466, 1518897, 25511272, 7035649, 1430, 1, 1, 1, 9, 407, 57799, 12833546, 1558435125, 18649337311, 915028347, 4862, 1

Offset: 0

Ilya Gutkovskiy, Aug 09 2017

This is the transpose of the array in A090182.

			G.f. of column k: A_k(x) = 1 + x + (k + 1)*x^2 + (k^3 + k^2 + 2*k + 1)*x^3 + (k^6 + k^5 + 2*k^4 + 3*k^3 + 3*k^2 + 3*k + 1)*x^4 + ...
Square array begins:
  1,   1,     1,       1,        1,         1,  ...
  1,   1,     1,       1,        1,         1,  ...
  1,   2,     3,       4,        5,         6,  ...
  1,   5,    17,      43,       89,       161,  ...
  1,  14,   171,    1252,     5885,     20466,  ...
  1,  42,  3113,  104098,  1518897,  12833546,  ...

Seiichi Manyama, Antidiagonals n = 0..55, flattened

Columns k=0-11 give: A000012, A000108, A015083, A015084, A015085, A015086, A015089,  A015091, A015092, A015093, A015095, A015096.
Main diagonal gives A290777.
Cf. A090182, A290789.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(j, k)*A(n-j-1, k)*k^j, j=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 10 2017

Table[Function[k, SeriesCoefficient[1/(1 - x/(1 + ContinuedFractionK[-k^i x, 1, {i, 1, n}])), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

from sympy.core.cache import cacheit
@cacheit
def A(n, k): return 1 if n==0 else sum(A(j, k)*A(n - j - 1, k)*k**j for j in range(n))
for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Aug 10 2017, after Maple code

G.f. of column k: 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))), a continued fraction.

A090182 Triangle T(n,k), 0 <= k <= n, composed of k-Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 17, 4, 1, 1, 1, 42, 171, 43, 5, 1, 1, 1, 132, 3113, 1252, 89, 6, 1, 1, 1, 429, 106419, 104098, 5885, 161, 7, 1, 1, 1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1, 1, 4862, 915028347, 18649337311, 1558435125, 12833546, 57799, 407, 9, 1, 1

Offset: 0

Philippe Deléham, Jan 20 2004, Oct 16 2008

			Triangle begins:
  1;
  1,    1;
  1,    1,       1;
  1,    2,       1,        1;
  1,    5,       3,        1,       1;
  1,   14,      17,        4,       1,     1;
  1,   42,     171,       43,       5,     1,   1;
  1,  132,    3113,     1252,      89,     6,   1, 1;
  1,  429,  106419,   104098,    5885,   161,   7, 1, 1;
  1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1;
This sequence formatted as a square array:
  1, 1, 1,   1,     1,        1,           1,               1, ...
  1, 1, 2,   5,    14,       42,         132,             429, ...
  1, 1, 3,  17,   171,     3113,      106419,         7035649, ...
  1, 1, 4,  43,  1252,   104098,    25511272,     18649337311, ...
  1, 1, 5,  89,  5885,  1518897,  1558435125,   6386478643785, ...
  1, 1, 6, 161, 20466, 12833546, 40130703276, 627122621447281, ...

Alois P. Heinz, Rows n = 0..55, flattened
Lun Lv, Zhihong Liu, Some Identities Related to Restricted Lattice Paths, 2016 9th International Symposium on Computational Intelligence and Design (ISCID), pp. 338-340.

The column sequences (without leading zeros) are A000012, A000108 (Catalan), A015083, A015084, A015085, A015086, A015089, A015091, A015092, A015093, A015095, A015096 for k=0..11.
T(2n,n) gives A290777.
Cf. A290759.

T:= proc(n, k) option remember; `if`(k=n, 1, add(
      T(j+k, k)*T(n-j-1, k)*k^j, j=0..n-k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Aug 10 2017

nmax = 10; col[k_] := col[k] = Module[{A}, A[] = 0; Do[A[x] = Normal[1/(1 - x*A[k*x]) + O[x]^(nmax-k+1)], {nmax-k+1}]; CoefficientList[A[x], x]];
T[n_, k_] := col[k][[n-k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2019, using g.f. given for column sequences *)

A125791 a(n) = 2^(n(n-1)(n-2)/6) for n>=1.

Original entry on oeis.org

1, 1, 2, 16, 1024, 1048576, 34359738368, 72057594037927936, 19342813113834066795298816, 1329227995784915872903807060280344576, 46768052394588893382517914646921056628989841375232

Offset: 1

Paul D. Hanna, Dec 10 2006

nonn

a(n) is a tetrahedral power of 2; exponents of 2 in a(n) begin: 0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ..., n*(n-1)*(n-2)/6, ... (cf. A000292).
Table A125790 is related to partitions into powers of 2, with A002577 in column 1 of A125790; further, column k of A125790 equals row sums of matrix power A078121^k, where triangle A078121 shifts left one column under matrix square.
Also number of distinct instances of the one-in-three monotone 3SAT problem for n variables. - Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008
Hankel transform of aerated 2-Catalan numbers (A015083). [Paul Barry, Dec 15 2010]

Michael De Vlieger, Table of n, a(n) for n = 1..28
Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.

Cf. A125790, A078121; A002577, A000292.

seq(2^(binomial(n, n-3)), n=1..10); # Zerinvary Lajos, Jun 16 2007 [modified by Georg Fischer, Nov 09 2023]

A125791[n_]:=2^Binomial[n,n-3];Array[A125791,15] (* Paolo Xausa, Nov 05 2023 *)

PARI
```
a(n)=if(n<1,0,2^(n*(n-1)*(n-2)/6))
```

/* As determinant of n X n matrix: */
{a(n)=local(q=2,A=Mat(1), B); for(m=1, n, B=matrix(m, m);
for(i=1, m, for(j=1, i, if(j==i||j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B);
return(matdet(matrix(n,n,r,c,(A^c)[r,1])))}
for(n=1,15,print1(a(n),", "))

% This generates all 3SAT problem instances
test:-test(4).
test(Max):-
between(1,Max,N),
nl,
one_in_three_monotone_3sat(N,Pss),
write(N:Pss),nl,
fail
; nl.
% generates all one-in-three monotone 3SAT problems involving N variables
one_in_three_monotone_3sat(N,Pss):-
ints(1,N,Is),
findall(Xs,ksubset(3,Is,Xs),Xss),
subset_of(Xss,Pss).
% subset generator
subset_of([],[]).
subset_of([X|Xs],Zs):-
subset_of(Xs,Ys),
add_element(X,Ys,Zs).
add_element(_,Ys,Ys).
add_element(X,Ys,[X|Ys]).
% subsets of K elements
ksubset(0,_,[]).
ksubset(K,[X|Xs],[X|Rs]):-K>0,K1 is K-1,ksubset(K1,Xs,Rs).
ksubset(K,[_|Xs],Rs):-K>0,ksubset(K,Xs,Rs).
% list of integers in [From..To]
ints(From,To,Is):-findall(I,between(From,To,I),Is).
% Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008

Determinant of n X n upper left corner submatrix of table A125790.

a(n) = 2^(binomial(n,n-3)). - Zerinvary Lajos, Jun 16 2007, modified to reflect the new offset by Paolo Xausa, Nov 06 2023.

Name simplified; determinant formula moved out of name into formula section by Paul D. Hanna, Oct 16 2013

Offset changed to 1 by Paolo Xausa, Nov 06 2023

A135867 G.f. satisfies A(x) = 1 + xA(2x)^2.

Original entry on oeis.org

1, 1, 4, 36, 640, 21888, 1451008, 188941312, 48768745472, 25069815595008, 25722272102744064, 52730972085034156032, 216091838647321476726784, 1770657164881170759078117376, 29013990909330956353981535748096

Offset: 0

Paul D. Hanna, Dec 02 2007

nonn

Self-convolution equals A135868 such that 2^n*A135868(n) = a(n+1) for n >= 0.

Seiichi Manyama, Table of n, a(n) for n = 0..81

Cf. A015083, A135868, A171192.

Cf. A171200, A171202, A171204, A171206, A171208, A171210, A343439.

nmax = 15; A[] = 0; Do[A[x] = 1 + x*A[2*x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 04 2021 *)

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,2*x)^2);polcoeff(A,n)}

a(n)=if(n==0,1,2^(n-1)*sum(k=0,n-1,a(k)*a(n-k-1))) \\ Paul D. Hanna, Feb 09 2010

a(n) = 2^(n-1)*Sum_{k=0..n-1} a(k)*a(n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Feb 09 2010

a(n) ~ c * 2^(n*(n+1)/2), where c = 0.715337433614869740944075474484711589980951273610257702786245519231799678... - Vaclav Kotesovec, Nov 04 2021

A171192 G.f. satisfies A(x) = 1/(1 - x*A(2x)^2).

Original entry on oeis.org

1, 1, 5, 53, 1045, 37941, 2596693, 343615093, 89402126741, 46139256172725, 47433024462021589, 97333484052884523765, 399068205440018335950357, 3270764880283567936326235445, 53601302478763156422575938811989

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..81

Cf. A015083, A171193-A171198.

nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^2) ); polcoeff(A, n)}

a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} 2^(i+j) * a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Nov 03 2021

a(n) ~ c * 2^(n*(n+1)/2), where c = 1.3216968146657309382653061124105846042506... - Vaclav Kotesovec, Nov 03 2021

A171198 G.f. A(x) satisfies A(x) = 1/(1 - xA(2x)^8).

Original entry on oeis.org

1, 1, 17, 689, 53777, 7805201, 2138582801, 1132509669905, 1178804946216209, 2433551908785577745, 10007244528797884954897, 82140401194398306308608785, 1347106337625031145913841134865, 44163564651481078406730693648713489

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..79

Cf. A015083, A171192-A171197.

nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^8) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^8) ); polcoeff(A, n)}

a(n) ~ c * 2^(n*(n+5)/2), where c = 0.265929653305627916979803234586945454418485... - Vaclav Kotesovec, Nov 03 2021

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_9>=0 and x_1+x_2+...+x_9=n-1} (1/2)^x_1 * Product_{k=1..9} a(x_k). - Seiichi Manyama, Jul 06 2025

A343439 G.f.: 1 + 2^0x/(1 + 2^1x/(1 + 2^2x/(1 + 2^3x/(1 + 2^4*x/(1 + ...))))).

Original entry on oeis.org

1, 1, -2, 12, -136, 2736, -99616, 6810816, -900563072, 234247256832, -120883821425152, 124271556482829312, -255006726559759042560, 1045529090595650037657600, -8569159507007490469146992640, 140431398588497630920722150113280, -4602217897540461023955069241211781120

Offset: 0

Seiichi Manyama, Apr 15 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..81

Cf. A015083, A168491, A216966.

a(n) = my(A=1+O(x)); for(i=1, n, A=1+2^(n-i)*x/A); polcoef(A, n);

a(n) = if(n<2, 1, -sum(k=1, n-1, 2^k*a(k)*a(n-k)));

G.f. satisfies: A(x) = 1 + x/A(2*x).
G.f.: 1/(Sum_{k>=0} A015083(k) * (-x)^k).
a(0) = a(1) = 1 and a(n) = -Sum_{k=1..n-1} 2^k*a(k)*a(n-k) for n > 1.
a(n) = (-2)^(n-1) * A015083(n-1) for n > 0.

A348857 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(2*x))).

Original entry on oeis.org

1, 2, 7, 44, 481, 9254, 326395, 21927776, 2874607189, 744650622170, 383510575423471, 393869218949592212, 807827718206737362889, 3311287802485779192925838, 27136007596894473408507305443, 444677773080105539125038867872456, 14572535437424416878539776253365375549

Offset: 0

Ilya Gutkovskiy, Nov 02 2021

nonn

Cf. A007317, A015083, A348858, A348859.

nmax = 16; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[2 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + Sum[2^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]

a(n) = 1 + Sum_{k=0..n-1} 2^k * a(k) * a(n-k-1).

a(n) ~ c * 2^(n*(n-1)/2), where c = 10.96416094535958612421479005398505892527943513193882801485045169159164... - Vaclav Kotesovec, Nov 02 2021

A348901 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(2*x)).

Original entry on oeis.org

1, 1, 5, 49, 893, 30649, 2030213, 264198625, 68180168717, 35046644401609, 35958357173552597, 73714882938928013809, 302083844634245306686685, 2475275541582550287356775001, 40559867144321249927245807932197, 1329146863668196853655964629931680001

Offset: 0

Ilya Gutkovskiy, Nov 03 2021

nonn

Counts lower triangular (0,1) matrices with 1's on the diagonal which cannot be decomposed in a nontrivial block diagonal fashion. For example, the third time is 5, counting the matrices [100,110,111], [100,110,011], [100,010,111], [100,110,101], [100,010,101]. There are 3 other 3x3 lower triangular (0,1) matrices with 1's on the diagonal; those others have block decompositions. - David Speyer, Jul 09 2025

Michael De Vlieger, Table of n, a(n) for n = 0..81
Johann Cigler, Hankel determinants of backward shifts of powers of q, arXiv:2407.05768 [math.CO], 2024. See p. 6.
Mare et al., Counting lower triangular 0-1-matrices with connected Coxeter permutation, MathOverflow, 2025.

Cf. A001003, A015083, A348188, A348902.

nmax = 15; A[] = 0; Do[A[x] = 1/(1 + x - 2 x A[2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[2^(k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]

a(0) = 1; a(n) = -a(n-1) + Sum_{k=0..n-1} 2^(k+1) * a(k) * a(n-k-1).
a(n) ~ 2^(n*(n+1)/2). - Vaclav Kotesovec, Nov 03 2021
G.f. A(x) satisfies 1/(1 - x*A(x)) = Sum_{n>=0} 2^(n(n-1)/2) * x^n. - David Speyer, Jul 09 2025

cp's OEIS Frontend

A015108 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-11.

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A290759 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))).

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A090182 Triangle T(n,k), 0 <= k <= n, composed of k-Catalan numbers.

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Maple

Mathematica

A125791 a(n) = 2^(n*(n-1)*(n-2)/6) for n>=1.

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Maple

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PARI

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A135867 G.f. satisfies A(x) = 1 + x*A(2*x)^2.

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Mathematica

PARI

PARI

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A171192 G.f. satisfies A(x) = 1/(1 - x*A(2x)^2).

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Mathematica

PARI

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A171198 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^8).

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A290759 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - kx/(1 - k^2x/(1 - k^3x/(1 - k^4x/(1 - ...)))))).

A125791 a(n) = 2^(n(n-1)(n-2)/6) for n>=1.

A135867 G.f. satisfies A(x) = 1 + xA(2x)^2.

A171198 G.f. A(x) satisfies A(x) = 1/(1 - xA(2x)^8).

A343439 G.f.: 1 + 2^0x/(1 + 2^1x/(1 + 2^2x/(1 + 2^3x/(1 + 2^4*x/(1 + ...))))).