A138364 -id:A138364 - cp's OEIS frontend

A348611 Number of ordered factorizations of n with no adjacent equal factors.

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 14, 1, 3, 3, 6, 1, 13, 1, 7, 3, 3, 3, 17, 1, 3, 3, 14, 1, 13, 1, 6, 6, 3, 1, 29, 1, 6, 3, 6, 1, 14, 3, 14, 3, 3, 1, 36, 1, 3, 6, 14, 3, 13, 1, 6, 3, 13, 1, 45, 1, 3, 6, 6, 3, 13, 1, 29, 4, 3

Offset: 1

Gus Wiseman, Nov 07 2021

nonn

First differs from A348610 at a(24) = 14, A348610(24) = 12.
An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
In analogy with Carlitz compositions, these may be called Carlitz ordered factorizations.

			The a(n) ordered factorizations without adjacent equal factors for n = 1, 6, 12, 16, 24, 30, 32, 36 are:
  ()   6     12      16      24      30      32      36
       2*3   2*6     2*8     3*8     5*6     4*8     4*9
       3*2   3*4     8*2     4*6     6*5     8*4     9*4
             4*3     2*4*2   6*4     10*3    16*2    12*3
             6*2             8*3     15*2    2*16    18*2
             2*3*2           12*2    2*15    2*8*2   2*18
                             2*12    3*10    4*2*4   3*12
                             2*3*4   2*3*5           2*3*6
                             2*4*3   2*5*3           2*6*3
                             2*6*2   3*2*5           2*9*2
                             3*2*4   3*5*2           3*2*6
                             3*4*2   5*2*3           3*4*3
                             4*2*3   5*3*2           3*6*2
                             4*3*2                   6*2*3
                                                     6*3*2
                                                     2*3*2*3
                                                     3*2*3*2
Thus, of total A074206(12) = 8 ordered factorizations of 12, only factorizations 2*2*3 and 3*2*2 (see A348616) are not included in this count, therefore a(12) = 6. - _Antti Karttunen_, Nov 12 2021

The additive version (compositions) is A003242, complement A261983.
The additive alternating version is A025047, ranked by A345167.
Factorizations without a permutation of this type are counted by A333487.
As compositions these are ranked by A333489, complement A348612.
Factorizations with a permutation of this type are counted by A335434.
The non-alternating additive version is A345195, ranked by A345169.
The alternating case is A348610, which is dominated at positions A122181.
The complement is counted by A348616.
A001055 counts factorizations, strict A045778, ordered A074206.
A325534 counts separable partitions, ranked by A335433.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A348613 counts non-alternating ordered factorizations.
Cf. A001250, A138364, A336103, A347050, A347438, A347463, A347466, A347706, A348379, A348383.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],antirunQ]],{n,100}]

A348611(n, e=0) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d!=e), s += A348611(n/d, d))); (s)); \\ Antti Karttunen, Nov 12 2021

a(n) = A074206(n) - A348616(n).

A014330 Exponential convolution of Catalan numbers with themselves.

Original entry on oeis.org

1, 2, 6, 22, 92, 424, 2108, 11134, 61748, 356296, 2123720, 13002840, 81417520, 519550880, 3369559864, 22161337742, 147544048324, 992923683912, 6746101933304, 46226667046360, 319199694771696, 2219445498261152, 15529758665102416, 109291258152550712

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A014333, A053175, A126869, A138364.

A014330:= func< n | (&+[Binomial(n,k)*Catalan(k)*Catalan(n-k): k in [0..n]]) >;
[A014330(n): n in [0..40]]; // G. C. Greubel, Jan 06 2023

Table[Sum[Binomial[n, k]*Binomial[2*k, k]/(k+1)*Binomial[2*n-2*k, n-k]/(n-k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Feb 25 2014 *)

A014330(n)=sum(k=0,n,binomial(n,k)*A000108(k)*A000108(n-k))  \\ M. F. Hasler, Jan 13 2012

def c(n): return catalan_number(n)
def A014330(n): return sum( binomial(n,k)*c(k)*c(n-k) for k in range(n+1))
[A014330(n) for n in range(41)] # G. C. Greubel, Jan 06 2023

From Vladeta Jovovic, Jan 01 2004: (Start)
E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x))^2.
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k, k)/(k+1)*binomial(2*n-2*k, n-k)/(n-k+1).
a(n) = 4^n*Sum_{k=0..n} (-4)^(-k)*binomial(n, k)*binomial(k, floor(k/2))*binomial(k+1, floor((k+1)/2)).
a(n) = binomial(2*n, n)/(n+1)*hypergeometric3F2([-n-1, -n, 1/2], [2, 1/2-n], -1). (End)
(n + 1)*(n + 2)*a(n) = 4*(3*n^2 + n - 1)*a(n - 1) - 32*(n - 1)^2*a(n - 2). - Vladeta Jovovic, Jul 15 2004
a(n) = Sum_{k=0..n} binomial(n,k)*A000108(k)*A000108(n-k). - Philippe Deléham, Aug 23 2006
a(n) = (4*A053175(n) - A053175(n+1)/4) / ((n+2)*2^n). - Mark van Hoeij, Jul 02 2010
G.f.: (1-6*x)*hypergeometric2F1([1/2, 1/2],[2],16*x^2/(4*x-1)^2)/(2*x*(4*x-1)) - x*(8*x-1)*hypergeometric2F1([3/2, 3/2],[3],16*x^2/(4*x-1)^2)/(4*x-1)^3 + 1/(2*x). - Mark van Hoeij, Oct 25 2011
E.g.f.: hypergeometric1F1([1/2],[2],4*x)^2, coinciding with the above given e.g.f. - Wolfdieter Lang, Jan 13 2012
a(n) ~ 8^(n+1) / (Pi*n^3). - Vaclav Kotesovec, Feb 25 2014

More terms from Vincenzo Librandi, Feb 27 2014

A266455 T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with no element 1 greater than its north neighbor modulo n and the upper left element equal to 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 0, 5, 2, 1, 3, 39, 101, 9, 1, 0, 357, 6152, 3417, 44, 1, 10, 3471, 442285, 2079765, 184881, 265, 1, 0, 35003, 34667512, 1548940233, 1362641028, 14216101, 1854, 1, 35, 362265, 2875874396, 1294905380013, 12959331192279, 1511484069811

Offset: 1

R. H. Hardin, Dec 29 2015

Table starts
.....1..........1.............1..............1.............1................1
.....0..........1.............0..............3.............0...............10
.....1..........5............39............357..........3471............35003
.....2........101..........6152.........442285......34667512.......2875874396
.....9.......3417.......2079765.....1548940233.1294905380013.1166706600769605
....44.....184881....1362641028.12959331192279
...265...14216101.1511484069811
..1854.1477674733
.14833

			Some solutions for n=3 k=4
..0..2..2..1....0..0..0..1....0..2..0..1....0..0..1..2....0..1..0..2
..2..1..1..0....2..2..0..1....0..1..2..1....2..2..1..1....2..1..2..1
..2..0..1..0....2..1..2..1....2..1..2..0....2..1..0..0....1..0..2..0

R. H. Hardin, Table of n, a(n) for n = 1..50

Column 1 is A000166(n-1).

Row 2 is A138364(n-1).

A349159 Numbers whose sum of prime indices is twice their alternating sum.

Original entry on oeis.org

1, 12, 63, 66, 112, 190, 255, 325, 408, 434, 468, 609, 805, 832, 931, 946, 1160, 1242, 1353, 1380, 1534, 1539, 1900, 2035, 2067, 2208, 2296, 2387, 2414, 2736, 3055, 3108, 3154, 3330, 3417, 3509, 3913, 4185, 4340, 4503, 4646, 4650, 4664, 4864, 5185, 5684, 5863

Offset: 1

Gus Wiseman, Nov 23 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their alternating sum.

			The terms and their prime indices begin:
     1: ()
    12: (2,1,1)
    63: (4,2,2)
    66: (5,2,1)
   112: (4,1,1,1,1)
   190: (8,3,1)
   255: (7,3,2)
   325: (6,3,3)
   408: (7,2,1,1,1)
   434: (11,4,1)
   468: (6,2,2,1,1)
   609: (10,4,2)
   805: (9,4,3)
   832: (6,1,1,1,1,1,1)
   931: (8,4,4)
   946: (14,5,1)
  1160: (10,3,1,1,1)

These partitions are counted by A000712 up to 0's.
An ordered version is A348614, negative A349154.
The negative version is A348617.
The reverse version is A349160, counted by A006330 up to 0's.
A025047 counts alternating or wiggly compositions, complement A345192.
A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
A034871, A097805, and A345197 count compositions by alternating sum.
A035363 = partitions with alt sum 0, ranked by A066207, complement A086543.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >= 0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344607 counts partitions with rev-alt sum >= 0, ranked by A344609.
A346697 adds up odd-indexed prime indices.
A346698 adds up even-indexed prime indices.
Cf. A000070, A000290, A001700, A028260, A045931, A120452, A195017, A241638, A257991, A257992, A325698, A345958, A349155.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[1000],Total[primeMS[#]]==2*ats[primeMS[#]]&]

A056239(a(n)) = 2*A316524(a(n)).

A346697(a(n)) = 3*A346698(a(n)).

A350139 Number of non-weakly alternating ordered factorizations of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 12, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 20, 0, 0, 0, 0, 0, 2, 0, 10, 0, 0, 0, 12, 0

Offset: 1

Gus Wiseman, Dec 24 2021

nonn

The first odd term is a(180) = 69, which has, for example, the non-weakly alternating ordered factorization 2*3*5*3*2.
An ordered factorization of n is a finite sequence of positive integers > 1 with product n. Ordered factorizations are counted by A074206.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

			The a(n) ordered factorizations for n = 24, 36, 48, 60:
  (2*3*4)  (2*3*6)    (2*3*8)    (2*5*6)
  (4*3*2)  (6*3*2)    (2*4*6)    (3*4*5)
           (2*3*3*2)  (6*4*2)    (5*4*3)
           (3*2*2*3)  (8*3*2)    (6*5*2)
                      (2*2*3*4)  (10*3*2)
                      (2*3*4*2)  (2*3*10)
                      (2*4*3*2)  (2*2*3*5)
                      (3*2*2*4)  (2*3*5*2)
                      (4*2*2*3)  (2*5*3*2)
                      (4*3*2*2)  (3*2*2*5)
                                 (5*2*2*3)
                                 (5*3*2*2)

Positions of nonzero terms are A122181.
The strong version for compositions is A345192, ranked by A345168.
The strong case is A348613, complement A348610.
The version for compositions is A349053, complement A349052.
As compositions with ones allowed these are ranked by A349057.
The complement is counted by A349059.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A025047 counts weakly alternating compositions, ranked by A345167.
A335434 counts separable factorizations, complement A333487.
A345164 counts alternating perms of prime factors, with twins A344606.
A345170 counts partitions with an alternating permutation.
A348379 counts factorizations w/ alternating perm, complement A348380.
A348611 counts anti-run ordered factorizations, complement A348616.
A349060 counts weakly alternating partitions, complement A349061.
Cf. A003242, A138364, A339846, A339890, A344604, A345194, A347050, A347438, A347463, A347706, A349054.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@facs[n],!whkQ[#]&&!whkQ[-#]&]],{n,100}]

a(2^n) = A349053(n).

A292629 a(n) = n! * [x^n] exp(nx)BesselI(1,2*x).

Original entry on oeis.org

0, 1, 4, 30, 304, 3885, 59976, 1085973, 22571200, 529712667, 13856212600, 399773871068, 12612288989664, 431948624278795, 15960564546516240, 632898895109081310, 26809122466181751552, 1208177444352064438155, 57719104861915100554200, 2913802658820378870546498, 154991214138728849712151200

Offset: 0

Ilya Gutkovskiy, Sep 20 2017

nonn

The n-th term of the n-th binomial transform of A138364.

Seiichi Manyama, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Main diagonal of A292628.

Cf. A138364, A186925.

Table[n!*SeriesCoefficient[E^(n*x)*BesselI[1,2*x],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 20 2017 *)

a(n) = polcoef((1+n*x+x^2)^n, n-1); \\ Michel Marcus, May 01 2019

a(n) = A292628(n,n).
a(n) ~ BesselI(1,2) * n^n. - Vaclav Kotesovec, Sep 20 2017
a(n) = [x^(n-1)] (1+n*x+x^2)^n = [x^(n+1)] (1+n*x+x^2)^n. - Seiichi Manyama, May 01 2019

A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

Original entry on oeis.org

0, 9, 130, 135, 141, 153, 177, 193, 225, 2052, 2059, 2062, 2069, 2074, 2079, 2089, 2098, 2103, 2109, 2129, 2146, 2151, 2157, 2169, 2209, 2242, 2247, 2253, 2265, 2289, 2369, 2434, 2439, 2445, 2457, 2481, 2529, 2561, 2689, 2818, 2823, 2829, 2841, 2865, 2913

Offset: 1

Gus Wiseman, Nov 22 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

			The terms and corresponding compositions begin:
     0: ()
     9: (3,1)
   130: (6,2)
   135: (5,1,1,1)
   141: (4,1,2,1)
   153: (3,1,3,1)
   177: (2,1,4,1)
   193: (1,6,1)
   225: (1,1,5,1)
  2052: (9,3)
  2059: (8,2,1,1)
  2062: (8,1,1,2)
  2069: (7,2,2,1)
  2074: (7,1,2,2)
  2079: (7,1,1,1,1,1)
  2089: (6,2,3,1)
  2098: (6,1,3,2)
  2103: (6,1,2,1,1,1)

These compositions are counted by A224274 up to 0's.
An unordered version is A348617, counted by A001523 up to 0's.
The positive version is A349153, unreversed A348614.
The unreversed version is A349154.
Positive unordered unreversed: A349159, counted by A000712 up to 0's.
A positive unordered version is A349160, counted by A006330 up to 0's.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- Heinz number is given by A333219.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Alternating compositions are ranked by A345167, complement A345168.

stc[n_]:=Differences[Prepend[ Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,1000],Total[stc[#]]==-2*sats[stc[#]]&]

A348616 Number of ordered factorizations of n with adjacent equal factors.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 6, 1, 0, 1, 2, 0, 0, 0, 9, 0, 0, 0, 9, 0, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 19, 1, 2, 0, 2, 0, 6, 0, 6, 0, 0, 0, 8, 0, 0, 2, 18, 0, 0, 0, 2, 0, 0, 0, 31, 0, 0, 2, 2, 0, 0, 0, 19, 4, 0, 0, 8, 0, 0

Offset: 1

Gus Wiseman, Nov 08 2021

nonn

First differs from A348613 at a(24) = 6, A348613(24) = 8.

An ordered factorization of n is a finite sequence of positive integers > 1 with product n.

			The a(n) ordered factorizations with at least one pair of adjacent equal factors for n = 12, 24, 36, 60:
   2*2*3    2*2*6      6*6        15*2*2
   3*2*2    6*2*2      2*2*9      2*2*15
            2*2*2*3    3*3*4      2*2*3*5
            2*2*3*2    4*3*3      2*2*5*3
            2*3*2*2    9*2*2      3*2*2*5
            3*2*2*2    2*2*3*3    3*5*2*2
                       2*3*3*2    5*2*2*3
                       3*2*2*3    5*3*2*2
                       3*3*2*2
See also examples in A348611.

Positions of 0's are A005117.
Positions of 4's appear to be A030514.
Positions of 2's appear to be A054753.
Positions of 1's appear to be A168363.
The additive version (compositions) is A261983, complement A003242.
Factorizations with a permutation of this type are counted by A333487.
Factorizations without a permutation of this type are counted by A335434.
The complement is counted by A348611.
As compositions these are ranked by A348612, complement A333489.
Dominated by A348613 (non-alternating ordered factorizations).
A001055 counts factorizations, strict A045778, ordered A074206.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A001250, A025047, A122181, A138364, A347050, A347463, A347706, A348379, A348380, A348382, A348610.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],!antirunQ[#]&]],{n,100}]

a(n) = A074206(n) - A348611(n).

A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

Original entry on oeis.org

0, 12, 160, 193, 195, 198, 204, 216, 240, 2304, 2561, 2563, 2566, 2572, 2584, 2608, 2656, 2752, 2944, 3074, 3077, 3079, 3082, 3085, 3087, 3092, 3097, 3099, 3102, 3112, 3121, 3123, 3126, 3132, 3152, 3169, 3171, 3174, 3180, 3192, 3232, 3265, 3267, 3270, 3276

Offset: 1

Gus Wiseman, Nov 21 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The terms and corresponding compositions begin:
       0: ()
      12: (1,3)
     160: (2,6)
     193: (1,6,1)
     195: (1,5,1,1)
     198: (1,4,1,2)
     204: (1,3,1,3)
     216: (1,2,1,4)
     240: (1,1,1,5)
    2304: (3,9)
    2561: (2,9,1)
    2563: (2,8,1,1)
    2566: (2,7,1,2)
    2572: (2,6,1,3)
    2584: (2,5,1,4)

These compositions are counted by A224274 up to 0's.
Except for 0, a subset of A345919.
The positive version is A348614, reverse A349153.
An unordered version is A348617, counted by A001523.
The reverse version is A349155.
A positive unordered version is A349159, counted by A000712 up to 0's.
A000346 = even-length compositions with alt sum != 0, complement A001700.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000984, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917, A349160.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Necklaces are ranked by A065609, dual A333764, reversed A333943.
- Alternating compositions are ranked by A345167, complement A345168.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Total[stc[#]]==-2*ats[stc[#]]&]

A014333 Three-fold exponential convolution of Catalan numbers with themselves.

Original entry on oeis.org

1, 3, 12, 57, 306, 1806, 11508, 78147, 559962, 4201038, 32792472, 264946446, 2206077804, 18860908644, 165050642736, 1474389557739, 13413397423482, 124030117316238, 1163661348170328, 11060842687616610, 106377560784576612, 1034009073326130876

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..930

Cf. A000108, A014330, A126869, A138364.

m:=40;
R:=PowerSeriesRing(Rationals(), m);
f:= func< x | (&+[(k+1-x)*x^(2*k)/(Factorial(k)*Factorial(k+1)): k in [0..m+2]]) >;
Coefficients(R!(Laplace( Exp(6*x)*( f(x) )^3 ))); // G. C. Greubel, Jan 06 2023

nmax = 20; CoefficientList[Series[E^(6*x)*(BesselI[0, 2*x] - BesselI[1, 2*x])^3, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 13 2017 *)

m=40
def f(x): return sum((k+1-x)*x^(2*k)/(factorial(k)*factorial(k+1)) for k in range(m+2))
def A014333_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(6*x)*( f(x) )^3 ).egf_to_ogf().list()
A014333_list(m) # G. C. Greubel, Jan 06 2023

E.g.f.: exp(6*x)*(BesselI(0,2*x) - BesselI(1,2*x))^3. - Ilya Gutkovskiy, Nov 01 2017
From Vaclav Kotesovec, Nov 13 2017: (Start)
Recurrence: (n+1)*(n+2)*(n+3)*a(n) = 4*(6*n^3 + 13*n^2 + 2*n - 3)*a(n-1) - 4*(n-1)*(44*n^2 - 16*n - 21)*a(n-2) + 192*(n-2)*(n-1)*(2*n - 3)*a(n-3).
a(n) ~ 2^(2*n) * 3^(n + 9/2) / (Pi^(3/2) * n^(9/2)). (End)

cp's OEIS Frontend

A348611 Number of ordered factorizations of n with no adjacent equal factors.

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A014330 Exponential convolution of Catalan numbers with themselves.

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A266455 T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with no element 1 greater than its north neighbor modulo n and the upper left element equal to 0.

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A349159 Numbers whose sum of prime indices is twice their alternating sum.

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A350139 Number of non-weakly alternating ordered factorizations of n.

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A292629 a(n) = n! * [x^n] exp(n*x)*BesselI(1,2*x).

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A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

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A348616 Number of ordered factorizations of n with adjacent equal factors.

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A292629 a(n) = n! * [x^n] exp(nx)BesselI(1,2*x).