A247287 -id:A247287 - cp's OEIS frontend

A247286 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k weak peaks.

1, 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 7, 4, 1, 1, 16, 17, 11, 5, 1, 1, 32, 41, 30, 16, 6, 1, 1, 64, 98, 82, 48, 22, 7, 1, 1, 128, 232, 220, 144, 72, 29, 8, 1, 1, 256, 544, 581, 423, 233, 103, 37, 9, 1, 1, 512, 1264, 1512, 1216, 738, 356, 142, 46, 10, 1

Offset: 0

Emeric Deutsch, Sep 14 2014

A weak peak of a Motzkin path is a vertex on the top of a hump. A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the Motzkin path u*duu*h*h*dd, where u=(1,1), h=(1,0), d(1,-1), has 4 weak peaks (shown by the stars).
Row n (n>=1) contains n entries.
Sum of entries in row n is the Motzkin number A001006(n).
Sum(k*T(n,k), 0<=k<=n) = A247287(n).

			Row 3 is 1,2,1 because the Motzkin paths hhh, hu*d, u*dh, and u*h*d have 0, 1, 1, and 2 weak peaks (shown by the stars).
Triangle starts:
1;
1;
1,1;
1,2,1;
1,4,3,1;
1,8,7,4,1;

Alois P. Heinz, Rows n = 0..141, flattened

Cf. A001006, A247287.

eq := G = 1+z*G+z^2*(G-1/(1-z)+t/(1-t*z))*G: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 14 do P[n] := sort(expand(coeff(Gser, z, n))) end do: 1; for n to 14 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t, c) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, expand(b(x-1, y-1, false, 0)*z^c+b(x-1, y, t,
      `if`(t, c+1, 0))+ b(x-1, y+1, true, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, false, 0)):
seq(T(n), n=0..14);  # Alois P. Heinz, Sep 14 2014

b[x_, y_, t_, c_] := b[x, y, t, c] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y-1, False, 0]*z^c + b[x-1, y, t, If[t, c+1, 0]] + b[x-1, y+1, True, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, False, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

The g.f. G(t,z) satisfies G = 1 + z*G + z^2*(G - 1/(1-z) + t/(1-t*z))*G.

A385641 Partial sums of A097893.

Original entry on oeis.org

1, 3, 8, 20, 51, 133, 356, 972, 2695, 7557, 21372, 60840, 174097, 500295, 1442720, 4172752, 12099411, 35161001, 102375400, 298586652, 872177273, 2551118623, 7471195500, 21904500500, 64286141881, 188844619563, 555216323396, 1633658183432, 4810340397375, 14173698242137

Offset: 0

Mélika Tebni, Aug 03 2025

Second partial sums of the central trinomial coefficients (A002426).
Third partial sums of A025178 (sequence starting 1, 0, 2, 4, 12, 32, 90 .... with offset 0).
For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).
For p Pythagorean prime (A002144), a(p) - 3 == 0 (mod p).
Sequences with g.f. (1-x)^k / sqrt(1-2*x-3*x^2): this sequence (k=-2), A097893 (k=-1), A002426 (k=0), A025178 (k=1), A024997 (k=2), A026083 (k=3). - Mélika Tebni, Aug 25 2025

Cf. A002144, A002145, A002426, A024997, A025178, A026083, A097893, A128014, A247287, A295112, A383527.

a := series(exp(x)*(BesselI(0, 2*x) + 2*int(BesselI(0, 2*x), x) + int(int(BesselI(0, 2*x), x), x)), x = 0, 30): seq(n!*coeff(a, x, n), n = 0 .. 29);

a(n) = sum(k=0, n, sum(i=0, k, sum(j=0, i, binomial(i, i-j)*binomial(j, i-j)))); \\ Michel Marcus, Aug 06 2025

from math import comb as C
def a(n):
    return sum(C(n+1, k+1)*C(2*(k//2), k//2) for k in range(n + 1))
print([a(n) for n in range(30)])

G.f.: (1 / sqrt((1 + x)*(1 - 3*x))) / (1 - x)^2.
E.g.f.: exp(x)*(BesselI(0, 2*x) + 2*g(x) + Integral_{x=-oo..oo} g(x) dx) where g(x) = Integral_{x=-oo..oo} BesselI(0, 2*x) dx.
D-finite with recurrence n*a(n) = (4*n-1)*a(n-1) - (2*n+1)*a(n-2) - (4*n-5)*a(n-3) + 3*(n-1)*a(n-4).
a(0) = 1, a(1) = 3 and a(n) = a(n-2) - 1 + 2*A383527(n) for n >= 2.
a(n) = Sum_{k=0..n} binomial(n+1, k+1)*A128014(k).
a(n) = Sum_{k=0..n} (2*A247287(k) + k+1).
a(n) ~ 3^(n + 5/2) / (8*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 03 2025
Sum_{k=0..n} A295112(n-k)*a(k) + binomial(n+3, 3) = 0. - Mélika Tebni, Sep 03 2025

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A247286 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k weak peaks.

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