A001924 -id:A001924 - cp's OEIS frontend

A104797 Triangle T(n,k) = Fib(n-k+4)-n-k-3, n>=1, 0<=k

1, 3, 1, 7, 3, 1, 14, 7, 3, 1, 26, 14, 7, 3, 1, 46, 26, 14, 7, 3, 1, 79, 46, 26, 14, 7, 3, 1, 133, 79, 46, 26, 14, 7, 3, 1, 221, 133, 79, 46, 26, 14, 7, 3, 1, 364, 221, 133, 79, 46, 26, 14, 7, 3, 1, 596, 364, 221, 133, 79, 46, 26, 14, 7, 3, 1, 972, 596, 364, 221, 133, 79, 46, 26

Offset: 1

Gary W. Adamson, Mar 26 2005

Repeatedly writing the sequence A001924 backwards.

			First few rows of the triangle are:
1;
3, 1;
7, 3, 1;
14, 7, 3, 1;
26, 14, 7, 3, 1;
46, 26, 14, 7, 3, 1;
...

Row sums are in A014162.

Cf. A104732.

Edited by Ralf Stephan, Apr 05 2009

A119997 Sum of all matrix elements of n X n matrix M[i,j] = (-1)^(i+j)*Fibonacci[i+j-1].

Original entry on oeis.org

1, 1, 4, 5, 17, 32, 97, 225, 628, 1573, 4225, 10880, 28769, 74849, 196708, 513765, 1347025, 3523360, 9229441, 24154625, 63251156, 165571781, 433507969, 1134881280, 2971250497, 7778684737, 20365103812, 53316141125, 139584105233, 365434903328, 956722661665

Offset: 1

Alexander Adamchuk, Aug 03 2006

Prime p divides a(p-1) for p={5,11,19,29,31,41,59,61,71,...} = A038872[n] Primes congruent to {0, 1, 4} mod 5. Also odd primes where 5 is a square mod p. p^2 divides a(p-1) for prime p={11,19,29,31,41,59,61,71,...} = A045468[n] Primes congruent to {1, 4} mod 5. Square prime divisors of a(n) up to n=50 are{2,3,5,7,11,13,19,23,29,31,41,47,89,101,139,151,199,211,461,521,3571,9349}, It appears that square prime divisors of a(n) belong to A061446[n] Primitive part of Fibonacci(n), A001578[n] Smallest primitive prime factor of Fibonacci number F(n) and A072183[n] Sequence arising from factorization of the Fibonacci numbers. Sum[Sum[Fibonacci[i+j-1],{i,1,n}],{j,1,n}] = A120297[n]. Sum[Sum[i+j-1,{i,1,n}],{j,1,n}] = n^3. Sum[Sum[(-1)^(i+j)*(i+j-1),{i,1,n}],{j,1,n}] = n for odd n and = 0 for even n.

			Matrix begins:
1 -1 2 -3 5
-1 2 -3 5 -8
2 -3 5 -8 13
-3 5 -8 13 -21
5 -8 13 -21 34

Colin Barker, Table of n, a(n) for n = 1..1000 [Terms up to n=200 from _Vincenzo Librandi_]
Index entries for linear recurrences with constant coefficients, signature (3,1,-7,5,-1).

Cf. A120297, A000045, A038872, A001924, A062381, A038872, A045468, A061446, A001578, A072183.

Table[Sum[Sum[(-1)^(i+j)*Fibonacci[i+j-1],{i,1,n}],{j,1,n}],{n,1,50}]

a(n) = sum(i=1, n, sum(j=1, n, (-1)^(i+j)*fibonacci(i+j-1))) \\ Colin Barker, Mar 26 2015

Vec(-x*(x^3+2*x-1)/((x-1)*(x^2-3*x+1)*(x^2-x-1)) + O(x^100)) \\ Colin Barker, Mar 26 2015

a(n) = Sum[Sum[(-1)^(i+j)*Fibonacci[i+j-1],{i,1,n}],{j,1,n}].
a(n) = 3*a(n-1)+a(n-2)-7*a(n-3)+5*a(n-4)-a(n-5) for n>5. - Colin Barker, Mar 26 2015
G.f.: -x*(x^3+2*x-1) / ((x-1)*(x^2-3*x+1)*(x^2-x-1)). - Colin Barker, Mar 26 2015

A131251 A000012 * A052509.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 3, 1, 5, 10, 7, 3, 1, 6, 15, 14, 7, 3, 1, 7, 21, 25, 15, 7, 3, 1, 8, 28, 41, 30, 15, 7, 3, 1, 9, 36, 63, 56, 31, 15, 7, 3, 1, 10, 45, 92, 98, 62, 31, 15, 7, 3, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A001924: (1, 3, 7, 14, 26, 46, 79, ...). A131252 = A052509 * A000012.
From Clark Kimberling, Feb 07 2011: (Start)
When formatted as a rectangle R with northwest corner
  1, 2, 3,  4,  5,  6, ...
  1, 3, 6, 10, 15, 21, ...
  1, 3, 7, 14, 25, 41, ...
  1, 3, 7, 15, 30, 56, ...
  1, 3, 7, 15, 31, 62, ...
  ...
the following properties hold:
R is the accumulation array of the transpose of A052553 (a version of Pascal's triangle); see A144112 for the definition of accumulation array.
row 1: A000027
row 2: A000217
row 3: A004006
row 4: A055795
row 5: A057703
row 6: A115567
limiting row:  A000225
antidiagonal sums: A001924.
(End)

			First few rows of the triangle:
  1;
  2,  1;
  3,  3,  1;
  4,  6,  3,  1;
  5, 10,  7,  3,  1;
  6, 15, 14,  7,  3,  1;
  7, 21, 25, 15,  7,  3,  1;
  ...

Cf. A052509, A000012, A001924, A131252.

A000012 * A052509 as infinite lower triangular matrices.

A136338 Primes in the array A136431 that are not Fibonacci numbers.

Original entry on oeis.org

7, 11, 29, 37, 41, 67, 79, 97, 137, 191, 211, 277, 379, 631, 709, 821, 947, 967, 991, 1129, 1327, 1597, 1831, 2017, 2081, 2267, 2347, 2557, 2683, 2851, 2927, 3571, 3917, 4561, 4657, 4951, 5051, 5779, 6217, 6329, 6763, 8273, 8647, 8779, 9181, 9871, 10093

Offset: 1

Jonathan Vos Post, Apr 12 2008

A generalization of prime Fibonacci numbers (A005478) are the prime hyperfibonacci numbers (primes in A136431). Referring to the array A(k,n) = Apply partial sum operator k times to Fibonacci numbers, we see that every prime occurs in the n=2 column (as it contains every positive integer).
So excluding n=2 and k=0 (A005478) we have the nontrivially prime hyperfibonacci numbers which are not Fibonacci numbers.
Note that this sequence does not indicate multiplicity (e.g., 7 occurs twice in the valid part of the table).
Continuing the table of primes in the examples, from a computation by Joshua Zucker, we have:
k=1: {7, ...} no more through n = 1000.
k=2: {7, 79, 514201, 14930317, 956722025983, 5527939700884681 4660046610375530219, ...}
k=3: {11, 97, 17519, next value has 60 digits, ...}
k=4: {41, 10093, 16703, 3520457, 591286703533, 6557470285501, 19740274219868101499, ...}
k=5: {709, 8273, 14323, 466004661037329684,1 298611126818977061133263, ...}
k=6: {29, 2683, 23945893, 1835540197, 4052735290427, 27777884012083, ...}
k=7: {37, 967, 2267, 127921, 226007, 62048869, 1131463777, 7540113804271826929, ...}
k=8: {27777538280521, 1409869790947669143312035590804646728957, ...}
k=9: {1033628323428189498226451492123369099, next value has 60 digits, ...}
k=10: {67, 5972304273877744135569337875802249660927, ...}
k=11: {79, 4478413, 19008291293, 61305228407581679, ...}
k=12: {6763, 1982269, 37886753582095837, 2791715456569622316696636389, ...}.

			k=1: primes in A000071 = {A000071(4) = 7}, no more through n = 1000.
k=2: primes in A001924 = {A001924(3) = 7, A001924(7) = 79, A001924(25) = 514201, ...}
k=3: primes in A014162 = {A014162(3) = 11, A014162(6) = 97, A014162(16) = 17519}, no more through n = 30.
k=4: primes in A014166 = {A014166(4) = 41, A014166(13) = 10093, A014166(14) = 16703}
k=5: primes in A053739 = {A053739(7) = 709, A053739(10) = 8273, A053739(11) = 14323}, no more through n = 27.
k=6: primes in A053295 = {A053295(3) = 29, A053295(8) = 2683, 23945893(24) = 23945893}, no more through n = 27.
k=7: primes in A053296 = {A053296(3) = 37, A053296(6) = 967, A053296(7) = 2267, A053296(12) = 127921, A053296(13) = 226007}, no more through n = 27.

Cf. A000040, A005478, A136431, A137176.

Cf. A136431.

A136431 := proc(k,n) local x ; coeftayl(x/(1-x-x^2)/(1-x)^k,x=0,n) ; end: A136338 := proc(amax) local a,k,n,a136431; a := [] ; for k from 1 do if A136431(k,3) > amax then break ; fi ; for n from 3 do a136431 := A136431(k,n) ; if a136431 > amax then break ; fi ; if isprime(a136431) and not a136431 in a then a := [op(a),a136431] ; fi ; od: od: sort(a) ; end: A136338(20000) ; # R. J. Mathar, Apr 21 2008

partsumfib(N,s=[],P=[])={ for( n=1+#s,N, s=concat(s,n+1); forstep( i=n,1,-1, isprime( s[i]+= if( i>1, s[i-1], fibonacci(n+2) ) ) & P=setunion(P,[s[i]]) ); print(s); );vecsort(eval(P))} \\ M. F. Hasler

Primes in the hyperfibonacci number array of A136431, excluding the n=2 column (which contains every positive integer).

Revised definition from N. J. A. Sloane, May 09 2008

More terms from R. J. Mathar, Apr 21 2008

A141289 Triangle read by rows, n-th row = (n-2)-th row appended to the beginning of (n-1)-th row, + n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 4, 5, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 4, 5, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 4, 5, 6, 7

Offset: 1

Gary W. Adamson, Jun 22 2008

There are (1, 2, 4, 7, 12,...) terms per row where (0, 0, 1, 2, 4, 7, 12,...) = A000071 = Fibonacci numbers - 1.

Row sums = A001924: (1, 3, 7, 14, 26, 46,...)

			First few rows of the triangle are:
1;
1, 2;
1, 1, 2, 3;
1, 2, 1, 1, 2, 3, 4;
1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 4, 5;
1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 4, 5, 6;
...
Row 4 = (1, 2, 1, 1, 2, 3, 4) = (row 2 appended to row 3, + 4); = (1, 2) appended to (1, 1, 2, 3), then 4.

Cf. A000071, A001924.

Triangle read by rows, n-th row = (n-2)-th row appended to the beginning of (n-1)-th row, + n.

A152688 Partial products of A152687.

Original entry on oeis.org

1, 1, 2, 48, 414720, 309586821120000, 62298599877271470735360000000000, 221419738218975714643056286355472083897548800000000000000000000, 30963454718960054822969246779894642673092903344400531870724683866888280945459200000000000000000000000000000000000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

nonn

Cf. A001924,A003266,A152686,A152687

lst={};p0=p1=p2=p3=1;Do[p0*=a[n];p1*=p0;p2*=p1;p3*=p2;AppendTo[lst,p3],{n,1,2*3!}];lst

Better definition from Omar E. Pol, Aug 06 2009

A210675 a(n)=a(n-1)+a(n-2)+n+4, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 7, 15, 30, 54, 94, 159, 265, 437, 716, 1168, 1900, 3085, 5003, 8107, 13130, 21258, 34410, 55691, 90125, 145841, 235992, 381860, 617880, 999769, 1617679, 2617479, 4235190, 6852702, 11087926, 17940663, 29028625, 46969325, 75997988, 122967352, 198965380

Offset: 0

Alex Ratushnyak, May 09 2012

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A210673: a(n)=a(n-1)+a(n-2)+n-4, a(0)=0,a(1)=1.
Cf. A066982: a(n)=a(n-1)+a(n-2)+n-2, a(0)=0,a(1)=1 (except the first term).
Cf. A104161: a(n)=a(n-1)+a(n-2)+n-1, a(0)=0,a(1)=1.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n,   a(0)=0,a(1)=1.
Cf. A192760: a(n)=a(n-1)+a(n-2)+n+1, a(0)=0,a(1)=1.
Cf. A192761: a(n)=a(n-1)+a(n-2)+n+2, a(0)=0,a(1)=1.
Cf. A192762: a(n)=a(n-1)+a(n-2)+n+3, a(0)=0,a(1)=1.

LinearRecurrence[{3, -2, -1, 1}, {0, 1, 7, 15}, 37] (* Jean-François Alcover, Oct 05 2017 *)

a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(4*x^2-4*x-1) / ((x-1)^2*(x^2+x-1)). - Colin Barker, May 31 2013

A210678 a(n) = a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 6, 12, 24, 43, 75, 127, 212, 350, 574, 937, 1525, 2477, 4018, 6512, 10548, 17079, 27647, 44747, 72416, 117186, 189626, 306837, 496489, 803353, 1299870, 2103252, 3403152, 5506435, 8909619, 14416087, 23325740, 37741862, 61067638, 98809537, 159877213, 258686789, 418564042

Offset: 0

Alex Ratushnyak, May 09 2012

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A081659: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=1 (except first 2 terms and sign).
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=1 (except first 4 terms).
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-2, a(0)=a(1)=1 (except first term).
Cf. A066982: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=1.
Cf. A030119: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=1.
Cf. A210677: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=1.

LinearRecurrence[{3,-2,-1,1},{1,1,6,12},40] (* Harvey P. Dale, Dec 10 2014 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+n+3}; NestList[nxt,{1,1,1},40][[;;,2]] (* Harvey P. Dale, Mar 19 2023 *)

From Colin Barker, Jun 30 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 -2*x + 5*x^2 - 3*x^3)/((1 - x)^2*(1 - x - x^2)). (End)

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.

A259454 Triangle T(n,k) (0 <= k <= n) read by rows, arising from the study of rook polynomials.

Original entry on oeis.org

1, 1, 3, 1, 6, 7, 1, 9, 22, 14, 1, 12, 46, 64, 26, 1, 15, 79, 177, 162, 46, 1, 18, 121, 380, 571, 374, 79, 1, 21, 172, 700, 1496, 1632, 809, 133, 1, 24, 232, 1164, 3261, 5116, 4270, 1668, 221, 1, 27, 301, 1799, 6271, 13013, 15754, 10446, 3316, 364

Offset: 0

N. J. A. Sloane, Jun 28 2015

See Riordan 1954 page 18 equation (9). - Michael Somos, Aug 26 2015

			Triangle T(n,k) begins:
1;
1,  3;
1,  6,  7;
1,  9,  22,   14;
1, 12,  46,   64,   26;
1, 15,  79,  177,  162,   46;
1, 18, 121,  380,  571,  374,   79;
1, 21, 172,  700, 1496, 1632,  809,  133;
1, 24, 232, 1164, 3261, 5116, 4270, 1668, 221;
G.f. = 1 + (1 + 3*t)*u + (1 + 6*t + 7*t^2)*u^2 + ...

Alois P. Heinz, Rows n = 0..140, flattened
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] (See triangle on page 18)

Some diagonals: A001924, A001925, A001926.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      T(n-1, k) +2*T(n-1, k-1) +T(n-2, k-1)
     -T(n-3, k-3) +`if`(n=k, 1, 0))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jul 02 2015

T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k] + 2*T[n-1, k-1] + T[n-2, k - 1] - T[n-3, k-3] + Boole[n == k]; T[, ] = 0; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2016 *)

{T(n, k) = polcoeff( polcoeff( 1 / ((1 - y*x) * (1 - (1 + 2*y)*x - y*x^2 + y^3*x^3)) + x * O(x^n), n), k)}; /* Michael Somos, Aug 26 2015 */

From Eq. (11) of Riordan (1954): T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) - T(n-3,k-3) + delta(n,k), where delta(n,k)=1 iff n=k, otherwise 0.

Sum_{n, k} T(n, k) * x^n*y^k = 1 / ((1 - y*x) * (1 - (1 + 2*y)*x - y*x^2 + y^3*x^3)). - Michael Somos, Aug 26 2015

More terms from Alois P. Heinz, Jul 02 2015

cp's OEIS Frontend

A104797 Triangle T(n,k) = Fib(n-k+4)-n-k-3, n>=1, 0<=k

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A119997 Sum of all matrix elements of n X n matrix M[i,j] = (-1)^(i+j)*Fibonacci[i+j-1].

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A131251 A000012 * A052509.

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A136338 Primes in the array A136431 that are not Fibonacci numbers.

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Maple

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A141289 Triangle read by rows, n-th row = (n-2)-th row appended to the beginning of (n-1)-th row, + n.

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A152688 Partial products of A152687.

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A210675 a(n)=a(n-1)+a(n-2)+n+4, a(0)=0, a(1)=1.

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A210678 a(n) = a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.

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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.

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