A391046 Triangle read by rows: T(n,k) = binomial(n+1,k+1) * binomial(3*n-2*k+1,k) / (n+1), 0<=k<=n.

1, 1, 1, 1, 5, 1, 1, 12, 15, 1, 1, 22, 72, 35, 1, 1, 35, 220, 300, 70, 1, 1, 51, 525, 1430, 990, 126, 1, 1, 70, 1071, 4900, 7007, 2772, 210, 1, 1, 92, 1960, 13566, 33320, 28028, 6864, 330, 1, 1, 117, 3312, 32340, 122094, 179928, 96096, 15444, 495, 1

Offset

0,5

Comments

T(n,k) = number of operator monomials M in a 4-ary associative algebra with a unary linear operator L, where M has n total operations and exactly k occurrences of L. E.g., the T(n,k) = 15 such operator monomials are L(L(abcd)), L(L(a)bcd), L(L(a))bcd, L(aL(b)cd), L(abL(c)d), L(abcL(d)), L(a)L(b)cd, L(a)bL(c)d, L(a)bcL(d), aL(L(b))cd, aL(b)L(c)d, aL(b)cL(d), abL(L(c))d, abL(c)L(d), abcL(L(d)).

T(n,k) = number of 231-avoiding permutations of {1,2,...,3n-2k+1} with k+1 decreasing runs, each run having length congruent to 1 modulo 3. E.g., the T(3,2) = 15 such permutations are 654123, 653124, 652134, 643125, 642135, 632145, 543126, 542136, 532146, 432156, 165423, 165324, 164325, 154326, 126543.

T(n,k) = number of labeled Schröder paths of semilength n avoiding DDDD and having exactly k total occurrences of H and DD where each descent (maximal sequence run of down steps) of length l >=2 is labeled by a (l-1)-subset of {1,2}. E.g., the T(3,2) = 15 such paths are UDHH, HUDH, HHUD, UU(DD)_{1}H, UU(DD)_{2}H, UHDH, HUU(DD)_{1}, HUU(DD)_{2}, HUHD, UHU(DD)_{1}, UHU(DD)_{2}, UUH(DD)_{1}, UUH(DD)_{2}, UHHD, UUU(DDD)_{1,2}.

T(n,k) = number of labeled Dyck paths of semilength n+1 with k occurrences of UU, where every non-terminal descent of length l>=2 is labeled by a composition of l-1 consisting of 3 nonnegative parts (the final descent is unlabeled). E.g., the T(3,2) = 15 such paths are

- UUUDUDDD, UUDUUDDD, UDUUUDDD;

- UUU(DD)_SUDD where S belongs to {(1,0,0), (0,1,0), (0,0,1)};

- UU(DD)_SUUDD where S belongs to {(1,0,0), (0,1,0), (0,0,1)};

- UUU(DDD)_SUD where S belongs to {(2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1), (0,1,1)}.

Links

Table of n, a(n) for n=0..54.

Yu Hin Au and Murray R. Bremner, A new generalization of the Narayana numbers inspired by linear operators on associative d-ary algebras, arXiv:2511.13671 [math.CO], 2025.

Formula

G.f. u satisfies u = (1+x*y*u)*(1+x*u*(1+x*y*u)^2).

Example

The initial rows of the triangle are
  [0] 1
  [1] 1,  1
  [2] 1,  5,    1
  [3] 1, 12,   15,     1
  [4] 1, 22,   72,    35,     1
  [5] 1, 35,  220,   300,    70,     1
  [6] 1, 51,  525,  1430,   990,   126,    1
  [7] 1, 70, 1071,  4900,  7007,  2772,  210,   1
  [8] 1, 92, 1960, 13566, 33320, 28028, 6864, 330, 1

Mathematica

T[n_, k_]:=Binomial[n+1, k+1]*Binomial[3n-2k+1, k]/(n+1); Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* James C. McMahon, Dec 07 2025 *)

Crossrefs

Cf. A001263, A391045, A391047, A391048.

Row sums A300048.

Sequence in context: A298328 A299221 A300035 * A381932 A130227 A114123

Adjacent sequences: A391043 A391044 A391045 * A391047 A391048 A391049

Keyword

easy,tabl,nonn

Author

Yu Hin Au, Nov 27 2025

Status

approved