A357368 -id:A357368 - cp's OEIS frontend

A237596 Convolution triangle of A000958(n+1).

1, 1, 1, 3, 2, 1, 8, 7, 3, 1, 24, 22, 12, 4, 1, 75, 73, 43, 18, 5, 1, 243, 246, 156, 72, 25, 6, 1, 808, 844, 564, 283, 110, 33, 7, 1, 2742, 2936, 2046, 1092, 465, 158, 42, 8, 1, 9458, 10334, 7449, 4178, 1906, 714, 217, 52, 9, 1, 33062, 36736, 27231, 15904, 7670, 3096, 1043, 288, 63, 10, 1

Offset: 0

Philippe Deléham, Feb 09 2014

Riordan array (f(x)/x, f(x)) where f(x) is the g.f. of A000958.
Reversal of A236918.
Row sums are A109262(n+1).
Diagonal sums are A033297(n+2).

			Triangle begins:
    1;
    1,   1;
    3,   2,   1;
    8,   7,   3,   1;
   24,  22,  12,   4,   1;
   75,  73,  43,  18,   5,  1;
  243, 246, 156,  72,  25,  6, 1;
  808, 844, 564, 283, 110, 33, 7, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.

Columns give A000958, A114495.

Cf. A033297 (diagonal sums), A109262 (row sums), A236918 (row reversal).

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> A000958(n)); # Peter Luschny, Oct 19 2022

P[n_, x_]:= P[n, x]= If[n==0, 1, Sum[(j/(2*n-j))*Binomial[2*n-j, n-j]*Fibonacci[j, x], {j,0,n}]];
T[n_, k_] := Coefficient[P[n+1, x], x, k];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 14 2022 *)

def f(n,x): return sum( binomial(n-j-1, j)*x^(n-2*j-1) for j in (0..(n-1)//2) )
def p(n,x):
    if (n==0): return 1
    else: return sum( (j/(2*n-j))*binomial(2*n-j, n-j)*f(j, x) for j in (0..n) )
def A237596(n,k): return ( p(n+1,x) ).series(x, n+1).list()[k]
flatten([[A237596(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2022

G.f. for the column k-1: ((1-sqrt(1-4*x))^k/(1+sqrt(1-4*x) + 2*x)^k)/x.
Sum_{k=0..n} T(n,k) = A109262(n+1).
From G. C. Greubel, Jun 14 2022: (Start)
T(n, k) = coefficient of [x^k]( p(n+1, x) ), where p(n, x) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n-j)*Fibonacci(j, x) with p(0, x) = 1 and Fibonacci(n, x) are the Fibonacci polynomials.
T(n, k) = A236918(n, n-k). (End)

A344557 Triangle read by rows, T(n, k) = 2^(n - k)*M(n, k, 1/2, 1/2), where M(n, k, x, y) is a generalized Motzkin recurrence. T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 13, 11, 3, 1, 57, 36, 18, 4, 1, 201, 165, 70, 26, 5, 1, 861, 646, 339, 116, 35, 6, 1, 3445, 2863, 1449, 595, 175, 45, 7, 1, 14897, 12104, 6692, 2744, 950, 248, 56, 8, 1, 63313, 53769, 29772, 13236, 4686, 1422, 336, 68, 9, 1

Offset: 0

Peter Luschny, May 25 2021

The convolution triangle of A091147. - Peter Luschny, Oct 07 2022

			[0]     1;
[1]     1,     1;
[2]     5,     2,     1;
[3]    13,    11,     3,     1;
[4]    57,    36,    18,     4,    1;
[5]   201,   165,    70,    26,    5,    1;
[6]   861,   646,   339,   116,   35,    6,   1;
[7]  3445,  2863,  1449,   595,  175,   45,   7,  1;
[8] 14897, 12104,  6692,  2744,  950,  248,  56,  8, 1;
[9] 63313, 53769, 29772, 13236, 4686, 1422, 336, 68, 9, 1.

A091147 (first column), A344558 (row sums).

t := proc(n, k) option remember; if n = k then 1 elif k < 0 or n < 0 or k > n then 0 elif k = 0 then t(n-1, 0)/2 + t(n-1, 1) else t(n-1, k-1) + (1/2)*t(n-1, k) + t(n-1, k+1) fi end: T := (n, k) -> 2^(n-k) * t(n, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Uses function PMatrix from A357368. Adds a row and column above and to the right.
PMatrix(10, n -> simplify(hypergeom([1/2-n/2, 1-n/2], [2], 16))); # Peter Luschny, Oct 07 2022

(* Uses function PMatrix from A357368 *)
nmax = 9;
M = PMatrix[nmax+2, HypergeometricPFQ[{1/2 - #/2, 1 - #/2}, {2}, 16]&];
T[n_, k_] := M[[n+2, k+2]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023, after Peter Luschny *)

The generalized Motzkin recurrence M(n, k, x, y) is defined as follows:

If k < 0 or n < 0 or k > n then 0 else if n = 0 then 1 else if k = 0 then x*M(n-1, 0, x, y) + M(n-1, 1, x, y). In all other cases M(n, k, x, y) = M(n-1, k-1, x, y) + y*M(n-1, k, x, y) + M(n-1, k+1, x, y).

A380115 a(n) = max{A380114(n, k) : k = 0..n}.

Original entry on oeis.org

1, 2, 4, 16, 48, 192, 640, 2560, 8960, 35840, 129024, 516096, 1892352, 7569408, 28114944, 112459776, 421724160, 1686896640, 6372720640, 25490882560, 96865353728, 387461414912, 1479398129664, 5917592518656, 22684104654848, 90736418619392, 348986225459200, 1395944901836800

Offset: 0

Peter Luschny, Feb 03 2025

nonn

These are the maxima of the rows of the convolution triangle of 2^n. The maxima of the convolution triangle of 2^(n-1) is A109388. For the definition of the convolution triangle see A357368.

Cf. A380114, A109388, A357368.

A162986 Number of Dyck paths with no UUU's and no DDD's of semilength n and having k UD's starting at level 0 (i.e., hills); (0 <= k <= n; U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 4, 5, 3, 4, 0, 1, 8, 10, 9, 4, 5, 0, 1, 17, 21, 18, 14, 5, 6, 0, 1, 37, 46, 40, 28, 20, 6, 7, 0, 1, 82, 102, 90, 66, 40, 27, 7, 8, 0, 1, 185, 230, 204, 152, 100, 54, 35, 8, 9, 0, 1, 423, 526, 469, 353, 235, 143, 70, 44, 9, 10, 0, 1, 978, 1216

Offset: 0

Emeric Deutsch, Oct 11 2009

T is the convolution triangle based on T(n,0) = A004148(n-1) (n >= 1). - Peter Luschny, Oct 19 2022

			T(5,2)=3 because we have (UD)(UD)UUDUDD, (UD)UUDUDD(UD), and UUDUDD(UD)(UD) (the hills are placed between parentheses).
Triangle starts:
  1;
  0, 1;
  1, 0, 1;
  1, 2, 0, 1;
  2, 2, 3, 0, 1;
  4, 5, 3, 4, 0, 1;

Cf. A004148, A162987.

g := ((1-z-z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^3: G := 1/(1-t*z-z^2-z^3*g): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
# Alternative based on a modified form of A004148:
# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
M004148 := n -> `if`(n<3, 2-n, hypergeom([(2-n)/2, (3-n)/2, (3-n)/2, (4-n)/2], [2, 2-n, 3-n], 16)):
PMatrix(10, n -> simplify(M004148(n))); # Peter Luschny, Oct 19 2022

G(t,z) = 1/(1 - tz - z^2 - z^3*g), where g = 1 + zg + z^2*g + z^3*g^2.
Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0) = A004148(n-1) (n>=1).
Sum_{k=0..n} k*T(n,k) = A162987(n).

A357586 Triangle read by rows. Convolution triangle of A002467 (number of permutations with fixed points).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 15, 9, 3, 1, 0, 76, 38, 15, 4, 1, 0, 455, 198, 70, 22, 5, 1, 0, 3186, 1182, 378, 112, 30, 6, 1, 0, 25487, 8115, 2274, 629, 165, 39, 7, 1, 0, 229384, 63266, 15439, 3840, 965, 230, 49, 8, 1, 0, 2293839, 554656, 117921, 25966, 6006, 1401, 308, 60, 9, 1

Offset: 0

Peter Luschny, Oct 09 2022

			[0] 1;
[1] 0,      1;
[2] 0,      1,     1;
[3] 0,      4,     2,     1;
[4] 0,     15,     9,     3,    1;
[5] 0,     76,    38,    15,    4,   1;
[6] 0,    455,   198,    70,   22,   5,   1;
[7] 0,   3186,  1182,   378,  112,  30,   6,  1;
[8] 0,  25487,  8115,  2274,  629, 165,  39,  7,  1;
[9] 0, 229384, 63266, 15439, 3840, 965, 230, 49,  8,  1;

Cf. A002467.

# Uses function PMatrix from A357368.
PMatrix(10, n -> simplify(GAMMA(n+1) - GAMMA(n+1, -1)*exp(-1)));

cp's OEIS Frontend

A237596 Convolution triangle of A000958(n+1).

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A344557 Triangle read by rows, T(n, k) = 2^(n - k)*M(n, k, 1/2, 1/2), where M(n, k, x, y) is a generalized Motzkin recurrence. T(n, k) for 0 <= k <= n.

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A380115 a(n) = max{A380114(n, k) : k = 0..n}.

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A162986 Number of Dyck paths with no UUU's and no DDD's of semilength n and having k UD's starting at level 0 (i.e., hills); (0 <= k <= n; U=(1,1), D=(1,-1)).

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Maple

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A357586 Triangle read by rows. Convolution triangle of A002467 (number of permutations with fixed points).

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Maple