A172061 -id:A172061 - cp's OEIS frontend

A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

1, 1, 0, 1, 1, 2, 1, 2, 4, 6, 1, 3, 7, 13, 22, 1, 4, 11, 24, 46, 80, 1, 5, 16, 40, 86, 166, 296, 1, 6, 22, 62, 148, 314, 610, 1106, 1, 7, 29, 91, 239, 553, 1163, 2269, 4166, 1, 8, 37, 128, 367, 920, 2083, 4352, 8518, 15792, 1, 9, 46, 174, 541, 1461, 3544, 7896

Offset: 0

Peter Luschny, Nov 14 2012

Notation: If a sequence id is starred then the offset and/or some terms are different. Starred terms indicate the variance.
Row sums: [A026641 ] [1,  1,  4,  13,  46,  166,   610]
--
T(j+2, 2) [A000124*] [1*, 2 , 4,   7,  11,  16,     22]
T(j+3, 3) [A003600*] [1*, 2*, 6,  13,  24,  40,     62]
--
T(j  , j) [A072547 ] [1,  0,  2,   6,  22,  80,    296]
T(j+1, j) [A026641 ] [1,  1,  4,  13,  46,  166,   610]
T(j+2, j) [A014300 ] [1,  2,  7,  24,  86,  314,  1163]
T(j+3, j) [A014301*] [1,  3, 11,  40, 148,  553,  2083]
T(j+4, j) [A172025 ] [1,  4, 16,  62, 239,  920,  3544]
T(j+5, j) [A172061 ] [1,  5, 22,  91, 367, 1461,  5776]
T(j+6, j) [A172062 ] [1,  6, 29, 128, 541, 2232,  9076]
T(j+7, j) [A172063 ] [1,  7, 37, 174, 771, 3300, 13820]
--
T(2j  ,j) [Central ] [1,  1,  7,  40, 239, 1461,  9076]
T(2j+1,j) [A183160 ] [1,  2, 11,  62, 367, 2232, 13820]
T(2j+2,j) [        ] [1,  3, 16,  91, 541, 3300, 20476]
T(2j+3,j) [A199033*] [1,  4, 22, 128, 771, 4744, 29618]

			Triangle begins as:
[n]|k->
[0] 1
[1] 1, 0
[2] 1, 1,  2
[3] 1, 2,  4,  6
[4] 1, 3,  7, 13,  22
[5] 1, 4, 11, 24,  46,  80
[6] 1, 5, 16, 40,  86, 166, 296
[7] 1, 6, 22, 62, 148, 314, 610, 1106.

G. C. Greubel, Rows n=0..100 of triangle, flattened

A201635 := proc(n,k) option remember; local j;
if n=k then (-1)^n*add(binomial(-n,j), j=0..n)
else add(A201635(n-1,j), j=0..k) fi end:
for n from 0 to 7 do seq(A(n,k), k=0..n) od;

T[n_, k_]:= T[n, k]= If[k==n, (-1)^n*Sum[Binomial[-n, j], {j, 0, n}], Sum[T[n-1, j], {j, 0, k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 27 2019 *)

{T(n,k) = if(k==n, (-1)^n*sum(j=0,n, binomial(-n,j)), sum(j=0,k, T(n-1,j)))};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 27 2019

@CachedFunction
def A201635(n, k):
    if n==k: return (-1)^n*add(binomial(-n, j) for j in (0..n))
    return add(A201635(n-1, j) for j in (0..k))
for n in (0..7) : [A201635(n, k) for k in (0..n)]

A247285 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n (n>=1) having k (0<=k<=n-1) upper interactions.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 7, 19, 14, 1, 1, 9, 36, 59, 26, 1, 1, 11, 58, 150, 162, 46, 1, 1, 13, 85, 300, 543, 408, 79, 1, 1, 15, 117, 523, 1335, 1771, 966, 133, 1, 1, 17, 154, 833, 2747, 5303, 5335, 2184, 221, 1, 1, 19, 196, 1244, 5031, 12792, 19272, 15099, 4767, 364, 1

Offset: 1

Emeric Deutsch, Sep 11 2014

An upper interaction in a Dyck path is an occurrence of a string d^k u^k for some k>=1; here u = (1,1) and d = (1,-1). For example, the Dyck path uu[d(du)u]dd has 2 upper interactions, shown between parentheses.
Number of entries in row n is n.
Sum of entries in row n is the Catalan number A000108(n).
Sum(k*T(n,k), k>=0) = A172061(n-2).
The statistic "number of lower interactions", mentioned in the Le Borgne reference is basically identical with the statistic "pyramid weight" of the Denise and Simion reference (see A091866 and the bottom of p. 8 of the Le Borgne reference).
T(n+1,n) = A001924(n) for n>=1. - Alois P. Heinz, Sep 11 2014

			Row 3 is 1,3,1. Indeed, the number of upper interactions in uuuddd, uududd, uuddud, uduudd, and ududud are 0, 1, 1, 1, and 2, respectively.
Triangle starts:
1;
1,1;
1,3,1;
1,5,7,1;
1,7,19,14,1;
1,9,36,59,26,1;

Alois P. Heinz, Rows n = 1..141, flattened
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
Y. Le Borgne, Counting upper interactions in Dyck paths, Sem. Lotharingien de Combinatoire, 54, 2006, Article B54f.

Cf. A000108, A001924, A172061, A091866.

q := u*t: s := ((1+t-2*q-sqrt((1-t)*(1-t-4*q+4*q^2)))*(1/2))/(t*(1-q)): Q := proc (x, n) options operator, arrow: product(1-q^k*x, k = 0 .. n-1) end proc: A := -t*add(((q-t)*s/(1-q))^n*q^(binomial(n+2, 2)-1)/(Q(q, n)*Q(q*t*s^2, n)), n = 0 .. 15)/add(((q-t)*s/(1-q))^n*q^binomial(n+2, 2)*(1-t*q^n*s)/(Q(q, n)*Q(q*t*s^2, n)*(1-q^n*s)*(1-q^(n+1)*s)), n = 0 .. 15): Aser := simplify(series(A, t = 0, 22)): for n to 16 do P[n] := sort(coeff(Aser, t, n)) end do: for n to 13 do seq(coeff(P[n], u, j), j = 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, max(0, c-1))*
     `if`(c>0, z, 1)+b(x-1, y-1, true, 1+`if`(t, c, 0)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n-1))(b(2*n, 0, false, 0)):
seq(T(n), n=1..15);  # Alois P. Heinz, Sep 11 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, Max[0, c-1]]*If[c>0, z, 1] + b[x-1, y-1, True, 1 + If[t, c, 0] ] ] ] ]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, n-1}]][b[2*n, 0, False, 0]]; Table[T[n], {n, 1, 25}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

The g.f. A(t,u), where t marks semilength and u marks upper interactions, is given in Proposition 2 of the Le Borgne reference. It is extremely complex; the Maple program follows it (blindly), except that the infinite sums have been replaced by summations from n=0 to n=15.

cp's OEIS Frontend

A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

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