A391048 Triangle read by rows: T(n,k) = binomial(n+1,k+1) * binomial(5*n-4*k+1,k) / (n+1), 0<=k<=n.

1, 1, 1, 1, 7, 1, 1, 18, 28, 1, 1, 34, 156, 84, 1, 1, 55, 510, 910, 210, 1, 1, 81, 1265, 4845, 4095, 462, 1, 1, 112, 2646, 17710, 33915, 15288, 924, 1, 1, 148, 4928, 51156, 177100, 189924, 49504, 1716, 1, 1, 189, 8436, 125664, 690606, 1381380, 895356, 143208, 3003, 1

Offset

0,5

Comments

T(n,k) = number of operator monomials M in a 6-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) = 28 such operator monomials are L(L(abcdef)), L(L(a)bcdef), L(L(a))bcdef, L(aL(b)cdef), L(abL(c)def), L(abcL(d)ef), L(abcdL(e)f), L(abcdeL(f)), L(a)L(b)cdef, L(a)bL(c)def, L(a)bcL(d)ef, L(a)bcdL(e)f, L(a)bcdeL(f), aL(L(b))cdef, aL(b)L(c)def, aL(b)cL(d)ef, aL(b)cdL(e)f, aL(b)cdeL(f), abL(L(c))def, abL(c)L(d)ef, abL(c)dL(e)f, abL(c)deL(f), abcL(L(d))ef, abcL(d)L(e)f, abcL(d)eL(f), abcdL(L(e))f, abcdL(e)L(f), abcdeL(L(f)).

T(n,k) = number of 231-avoiding permutations of {1,2,...,5n-4k+1} with k+1 decreasing runs, each run having length congruent to 1 modulo 5. E.g., the T(3,2) = 28 such permutations are 87654123, 87653124, 87652134, 87643125, 87642135, 87632145, 87543126, 87542136, 87532146, 87432156, 86543127, 86542137, 86532147, 86432157, 85432167, 76543128, 76542138, 76532148, 76432158, 75432168, 65432178, 18765423, 18765324, 18764325, 18754326, 18654327, 17654328, 12876543.

T(n,k) = number of labeled Schröder paths of semilength n avoiding DDDDDD 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 {1,2,3,4}. E.g., the T(3,2) = 28 such paths are

- UDHH, HUDH, HHUD, UHDH, HUHD, UHHD;

- UU(DD)_SH where S belongs to {{1}, {2}, {3}, {4}};

- HUU(DD)_S where S belongs to {{1}, {2}, {3}, {4}};

- UHU(DD)_S where S belongs to {{1}, {2}, {3}, {4}};

- UUH(DD)_S where S belongs to {{1}, {2}, {3}, {4}};

- UUU(DDD)_S where S belongs to {{1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}.

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 5 nonnegative parts (the final descent is unlabeled). E.g., the T(3,2) = 28 such paths are

- UUUDUDDD, UUDUUDDD, UDUUUDDD;

- UUU(DD)_SUDD where S belongs to {(1,0,0,0,0), ..., (0,0,0,0,1)} (5 choices);

- UU(DD)_SUUDD where S belongs to {(1,0,0,0,0), ..., (0,0,0,0,1)} (5 choices);

- UUU(DDD)_SUD where S belongs to {(2,0,0,0,0), ..., (0,0,0,1,1)} (15 choices).

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)^4).

Example

The initial rows of the triangle are
  [0] 1
  [1] 1,   1
  [2] 1,   7,    1
  [3] 1,  18,   28,     1
  [4] 1,  34,  156,    84,      1
  [5] 1,  55,  510,   910,    210,      1
  [6] 1,  81, 1265,  4845,   4095,    462,     1
  [7] 1, 112, 2646, 17710,  33915,  15288,   924,    1
  [8] 1, 148, 4928, 51156, 177100, 189924, 49504, 1716, 1

Mathematica

T[n_, k_]:=Binomial[n+1, k+1]*Binomial[5n-4k+1, k]/(n+1); Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* James C. McMahon, Dec 08 2025 *)

Crossrefs

Cf. A001263, A391045, A391046, A391047.

Row sums A364734.

Sequence in context: A141597 A372066 A176561 * A176284 A154233 A382740

Adjacent sequences: A391045 A391046 A391047 * A391049 A391050 A391051

Keyword

easy,tabl,nonn

Author

Yu Hin Au, Nov 27 2025

Status

approved