A001629 -id:A001629 - cp's OEIS frontend

A385817 Irregular triangle read by rows listing the lengths of maximal runs (sequences of consecutive elements increasing by 1) of binary indices, duplicate rows removed.

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

Offset: 0

Gus Wiseman, Jul 14 2025

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

This is the triangle A245563, except all duplicates after the first instance of each composition are removed. It lists all compositions in order of their first appearance as a row of A245563.

			The binary indices of 53 are {1,3,5,6}, with maximal runs ((1),(3),(5,6)), with lengths (1,1,2). After removing duplicates, this is our row 16.
Triangle begins:
   0: .
   1: 1
   2: 2
   3: 1 1
   4: 3
   5: 2 1
   6: 1 2
   7: 4
   8: 1 1 1
   9: 3 1
  10: 2 2
  11: 1 3
  12: 5
  13: 2 1 1
  14: 1 2 1
  15: 4 1
  16: 1 1 2
  17: 3 2
  18: 2 3
  19: 1 4
  20: 6
  21: 1 1 1 1

In the following references, "before" is short for "before removing duplicate rows".
Positions of singleton rows appear to be A000071 = A000045-1, before A023758.
Positions of firsts appearances appear to be A001629.
Positions of rows of the form (1,1,...) appear to be A055588 = A001906+1.
First term of each row appears to be A083368.
Row sums appear to be A200648, before A000120.
Row lengths after the first row appear to be A200650+1, before A069010 = A037800+1.
Before the removals we had A245563 (except first term), see A245562, A246029, A328592.
For anti-run ranks we have A385816, before A348366, firsts A052499.
Standard composition numbers of rows are A385818, before A385889.
For anti-runs we have A385886, before A384877, firsts A384878.
Cf. A044813, A048793, A164707, A200649, A242882, A243815, A384175, A384890.

DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2==#1+1&],{n,0,100}]]

A083368 a(n) is the position of the highest one in A003754(n).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 1, 3, 2, 1, 5, 2, 1, 4, 1, 3, 2, 1, 6, 1, 3, 2, 1, 5, 2, 1, 4, 1, 3, 2, 1, 7, 2, 1, 4, 1, 3, 2, 1, 6, 1, 3, 2, 1, 5, 2, 1, 4, 1, 3, 2, 1, 8, 1, 3, 2, 1, 5, 2, 1, 4, 1, 3, 2, 1, 7, 2, 1, 4, 1, 3, 2, 1, 6, 1, 3, 2, 1, 5, 2, 1, 4, 1, 3, 2, 1, 9, 2, 1, 4, 1, 3, 2, 1, 6, 1, 3, 2, 1, 5, 2

Offset: 1

Gary W. Adamson, Jun 04 2003

Previous name was: A Fibbinary system represents a number as a sum of distinct Fibonacci numbers (instead of distinct powers of two). Using representations without adjacent zeros, a(n) = the highest bit-position which changes going from n-1 to n.
A003754(n), when written in binary, is the representation of n.
Often one uses Fibbinary representations without adjacent ones (the Zeckendorf expansion).
a(A000071(n+3)) = n. - Reinhard Zumkeller, Aug 10 2014
From Gus Wiseman, Jul 24 2025: (Start)
Conjecture: To obtain this sequence, start with A245563 (maximal run lengths of binary indices), then remove empty and duplicate rows (giving A385817), then take the first term of each remaining row. Some variations:
- For sum instead of first term we appear to have A200648.
- For length instead of first term we appear to have A200650+1.
- For last instead of first term we have A385892.
(End)

			27 is represented 110111, 28 is 111010; the fourth position changes, so a(28)=4.

Jay Kappraff, Beyond Measure: A Guided Tour Through Nature, Myth and Number, World Scientific, 2002, page 460.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

A035612 is the analogous sequence for Zeckendorf representations.
A001511 is the analogous sequence for power-of-two representations.
Cf. A003714, A003754.
Cf. A000045, A000071.
Appears to be the first element of each row of A385817, see A083368, A200648, A200650, A385818, A385892.
A000120 counts ones in binary expansion.
A245563 gives run lengths of binary indices, see A089309, A090996, A245562, A246029, A328592.
A384877 gives anti-run lengths of binary indices, ranks A385816.
Cf. A001629, A037800, A044813, A048793, A069010, A200649, A384878, A385886.

a083368 n = a083368_list !! (n-1)
a083368_list = concat $ h $ drop 2 a000071_list where
   h (a:fs@(a':_)) = (map (a035612 . (a' -)) [a .. a' - 1]) : h fs
-- Reinhard Zumkeller, Aug 10 2014

For n = F(a)-1 to F(a+1)-2, a(n) = A035612(F(a+1)-1-n).

a(n) = a(k)+1 if n = ceiling(phi*k) where phi is the golden ratio; otherwise a(n) = 1. - Tom Edgar, Aug 25 2015

Edited by Don Reble, Nov 12 2005

Shorter name from Joerg Arndt, Jul 27 2025

A123585 Triangle T(n,k), 0<=k<=n, given by [1, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, -1, 1, 5, 3, -1, -2, 4, 10, 5, 0, -4, -4, 12, 20, 8, 1, -2, -13, -4, 31, 38, 13, 1, 3, -11, -33, 3, 73, 71, 21, 0, 6, 6, -42, -74, 34, 162, 130, 34, -1, 3, 24, 0, -130, -146, 128, 344, 235, 55, -1, -4, 21, 72, -50, -352

Offset: 0

Philippe Deléham, Nov 13 2006

			Triangle begins:
1;
1, 1;
0, 2, 2;
-1, 1, 5, 3;
-1, -2, 4, 10, 5;
0, -4, -4, 12, 20, 8;
1, -2, -13, -4, 31, 38, 13;
1, 3, -11, -33, 3, 73, 71, 21;
0, 6, 6, -42, -74, 34, 162, 130, 34;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A010892, A099254, A000045, A000079 (row sums), A001629, A129707.

CoefficientList[CoefficientList[Series[1/(1 - (1 + y)*x + (1 - y^2)*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 16 2017 *)

Sum_{k,0<=k<=n} T(n,k) = 2^n = A000079(n).
T(n,0) = A010892(n).
T(n,n) = Fibonacci(n+1) = A000045(n+1).
T(n+1,1) = A099254(n).
T(n+1,n) = A001629(n+2).
Sum_{k, 0<=k<=[n/2]} T(n-k,k) = A003269(n).
T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k), n>0.
Sum_{k, 0<=k<=n} x^k*T(n,k) = (-1)^n*A003683(n+1), (-1)^n*A006130(n), A000007(n), A010892(n), A000079(n), A030195(n+1) for x=-3, -2, -1, 0, 1, 2 respectively . - Philippe Deléham, Dec 01 2006
T(n+2,n) = A129707(n+1).- Philippe Deléham, Dec 18 2011
G.f.: 1/(1-(1+y)*x+(1-y^2)*x^2). - Philippe Deléham, Dec 18 2011

A246177 Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) such that the area between the x-axis and the path is k.

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 2, 1, 8, 5, 3, 1, 13, 10, 8, 4, 2, 21, 20, 18, 12, 7, 3, 1, 34, 38, 39, 30, 22, 12, 7, 2, 1, 55, 71, 80, 70, 57, 39, 26, 14, 7, 3, 1, 89, 130, 160, 154, 138, 106, 81, 52, 34, 18, 10, 4, 2, 144, 235, 312, 327, 315, 267, 220, 163, 118, 78, 49, 28, 16, 7, 3, 1

Offset: 0

Emeric Deutsch, Aug 20 2014

The members of B(n) are paths of weight n that start at (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
Apparently, number of terms in row n is 1+floor(n^2/8).
Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
T(n,0) = A000045(n+1) (the Fibonacci numbers).
T(n,1) = A001629(n-1) (n>=1).

			Row 3 is 3,1; indeed, B(3) consists of the paths hhh, hH, Hh, UD with areas 0,0,0,1, respectively.
Triangle starts:
   1;
   1;
   2;
   3,   1;
   5,   2,   1;
   8,   5,   3,   1;
  13,  10,   8,   4,   2;
  21,  20,  18,  12,   7,   3,  1;
  34,  38,  39,  30,  22,  12,  7,  2,  1;
  55,  71,  80,  70,  57,  39, 26, 14,  7,  3,  1;
  89, 130, 160, 154, 138, 106, 81, 52, 34, 18, 10, 4, 2;

Alois P. Heinz, Rows n = 0..65, flattened
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

Cf. A004148, A000045, A001629.

g := 1/(1-z-z^2-t*z^3*A[1]): for j to 15 do A[j] := 1/(1-t^j*z-t^j*z^2-t^(2*j+1)*z^3*A[j+1]) end do: gser := simplify(series(g, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, j), j = 0 .. floor((1/8)*n^2)) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y) option remember; `if`(y<0 or y>n, 0, `if`(n=0, 1,
      expand(b(n-1, y)*x^y +`if`(n>1, b(n-2, y)*x^y+b(n-2, y+1)*
      x^(y+1/2), 0) +b(n-1, y-1)*x^(y-1/2))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, Aug 20 2014

b[n_, y_] := b[n, y] = If[y<0 || y>n, 0, If[n == 0, 1, Expand[b[n-1, y]*x^y + If[n>1, b[n-2, y]*x^y + b[n-2, y+1]*x^(y+1/2), 0] + b[n-1, y-1]*x^(y-1/2)]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

The trivariate g.f. G=G(t,s,z), where t marks area, s marks length (=number of steps), and z marks weight, satisfies G = 1+szG+sz^2G+ts^2z^3G(t,ts,z)G. This follows at once from the fact that every nonempty path is of the form hC or HC or UCDC, where h denotes a (1,0)-step of weight 1, H denotes a (1,0)-step of weight 2, U denotes a (1,1)-step, D denotes a (1,-1)-step, and the C's denote paths, not necessarily the same. From the equation one can find G(t,s,z) as a continued fraction (the Maple program makes use of this).

A332724 Number of length n - 1 ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.

Original entry on oeis.org

0, 0, 1, 6, 14, 32, 65, 128, 243, 452, 826, 1490, 2659, 4704, 8261, 14418, 25030, 43252, 74437, 127648, 218199, 371920, 632306, 1072486, 1815239, 3066432, 5170825, 8705118, 14632958, 24562952, 41177801, 68947520, 115313979, 192656924, 321554986, 536191418

Offset: 0

Gus Wiseman, Mar 03 2020

nonn

In other words, parts of not-immediately-subsequent blocks are increasing.

			The a(2) = 1 through a(4) = 14 ordered set partitions:
  {{1,2}}  {{1},{2,3}}  {{1},{2},{3,4}}
           {{1,2},{3}}  {{1},{2,3},{4}}
           {{1,3},{2}}  {{1,2},{3},{4}}
           {{2},{1,3}}  {{1},{2,4},{3}}
           {{2,3},{1}}  {{1,2},{4},{3}}
           {{3},{1,2}}  {{1},{3},{2,4}}
                        {{1,3},{2},{4}}
                        {{1},{3,4},{2}}
                        {{1},{4},{2,3}}
                        {{2},{1},{3,4}}
                        {{2},{1,3},{4}}
                        {{2},{1,4},{3}}
                        {{2,3},{1},{4}}
                        {{3},{1,2},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Column k = n - 1 of A332673, which has row-sums A332872.
Ordered set-partitions are A000670.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal normal sequences are A328509.
Cf. A000045, A001286, A001629, A056242, A227038, A332577.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==n-1&&!MatchQ[#,{_,{_,a_,_},,{_,b_,_},_}/;a>b]&]],{n,0,8}]

\\ here b(n) is A001629(n).
b(n) = {((n+1)*fibonacci(n-1) + (n-1)*fibonacci(n+1))/5}
a(n) = {if(n==0, 0, b(n) + 4*b(n-1) + b(n-2))} \\ Andrew Howroyd, Apr 17 2021

From Andrew Howroyd, Apr 17 2021: (Start)
a(n) = A001629(n) + 4*A001629(n+1) + A001629(n+2) for n > 0.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n > 4.
G.f.: x*(1 + 4*x + x^2)/(1 - x - x^2)^2.
(End)

Terms a(9) and beyond from Andrew Howroyd, Apr 17 2021

A039913 Triangular "Fibonacci array".

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 5, 4, 5, 3, 5, 8, 7, 7, 8, 5, 8, 13, 11, 12, 11, 13, 8, 13, 21, 18, 19, 19, 18, 21, 13, 21, 34, 29, 31, 30, 31, 29, 34, 21, 34, 55, 47, 50, 49, 49, 50, 47, 55, 34, 55, 89, 76, 81, 79, 80, 79, 81, 76, 89, 55, 89, 144, 123, 131, 128, 129, 129, 128

Offset: 0

N. J. A. Sloane

Sum of n-th row = 2*A001629(n+1). - Reinhard Zumkeller, Oct 07 2012

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
L. Carlitz, A Fibonacci array, Fib. Quart. 1(#2) (1963), 217-28.

Cf. A108035.

a039913 n k = a039913_tabl !! n !! k
a039913_row n = a039913_tabl !! n
a039913_tabl = [[0], [1, 1]] ++ f [0] [1, 1] where
   f us@(u:us') vs@(v:vs') = ws : f vs ws where
     ws = [u + v, u + v + v] ++ zipWith (+) us vs'
-- Reinhard Zumkeller, Oct 07 2012

a(0, n)=Fib(n), a(1, n)=Fib(n+2), a(r, n)=a(r-1, n)+a(r-2, n), r >= 2.

G.f.: (x+y)/((1-x-x^2)*(1-y-y^2)). [U coordinates]

More terms from Larry Reeves (larryr(AT)acm.org), Sep 28 2000

A091562 Triangle read by rows, related to Pascal's triangle, starting with 1, 0, 0.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 5, 7, 5, 2, 3, 10, 17, 17, 10, 3, 5, 20, 41, 51, 41, 20, 5, 8, 38, 91, 136, 136, 91, 38, 8, 13, 71, 195, 339, 405, 339, 195, 71, 13, 21, 130, 403, 799, 1107, 1107, 799, 403, 130, 21, 34, 235, 812, 1807, 2845, 3297, 2845, 1807, 812, 235, 34

Offset: 0

Christian G. Bower, Jan 20 2004

			Triangle begins:
  1;
  0,0;
  1,1,1;
  1,2,2,1;
  2,5,7,5,2;
  ...

Row sums: A054878, column 0: A000045(n-1), column 1: A001629.
Cf. A090171, A090172, A090173, A090174, A091533, A205575 (same recurrence).
Cf. A090172.

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.

G.f.: A(x, y) = (1-x-x*y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).

A128502 Convolution array for Chebyshev's S(n,x)=U(n,x/2) polynomials.

Original entry on oeis.org

1, 2, 3, -2, 4, -6, 5, -12, 3, 6, -20, 12, 7, -30, 30, -4, 8, -42, 60, -20, 9, -56, 105, -60, 5, 10, -72, 168, -140, 30, 11, -90, 252, -280, 105, -6, 12, -110, 360, -504, 280, -42, 13, -132, 495, -840, 630, -168, 7, 14, -156, 660, -1320, 1260, -504, 56, 15, -182, 858, -1980, 2310, -1260, 252, -8, 16, -210, 1092

Offset: 0

Wolfdieter Lang Apr 04 2007

S1(n,x):=sum(S(n-k,x)*S(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)).
The unsigned column sequences, m>=0, divided by (m+1) give Pascal triangle column sequences for m+1.
G.f. for column m sequence: ((-1)^m)*(m+1)*(x^(2*m))/(1-x)^(m+2), m>=0.
Row polynomials P1(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).
Written as a triangle with increasing powers of x this is A294519. - Wolfdieter Lang, Nov 12 2017

			[1];[2];[3,-2],[4,-6];[5,-12,3];[6,-20,12];[7,-30,30,-4];[8,-42,60,-20];...
n=4: [5,-12,3] stands for the polynomial S1(4,x) = 5*x^4-12*x^2+3 = 2*(S(4,x)*1+S(3,x)*S(1,x))+S(2,x)*S(2,x).
n=4: [5,-12,3] stands also for the row polynomial P1(4,x) = 5-12*x+3*x^2.

W. Lang, First 15 rows and more.

Row sums (signed array) give A099254. Unsigned row sums are A001629(n+2).
Cf. A115139 (with offset n>=0 is S(n, x) array, decreasing powers of x).
Cf. A294519 (as triangle).

a(n,m)=binomial(n-m,m)*(n+1-m)*(-1)^m, m=0..floor(n/2), n>=0.
a(n,m)=binomial(n+1-m,m+1)*(m+1)*(-1)^m, m=0..floor(n/2), n>=0.
G.f. for S1(n,x): 1/(1-x*z+z^2)^2.
G.f. for P1(n,x): 1/(1-z+x*z^2)^2.

A191830 Expansion of x^2(2-3x)/(1-x-x^2)^2.

Original entry on oeis.org

0, 0, 2, 1, 4, 5, 10, 16, 28, 47, 80, 135, 228, 384, 646, 1085, 1820, 3049, 5102, 8528, 14240, 23755, 39592, 65931, 109704, 182400, 303050, 503161, 834868, 1384397, 2294290, 3800080, 6290788, 10408679, 17213696, 28454415, 47014380, 77647104, 128186062

Offset: 0

Paul Curtz, Jun 17 2011

a(2*n) mod 2 = 0;
a(4*n) mod 4 = 0;
a(5*n) mod 5 = 0 and a(5*n+1) mod 5 = 0;
a(n) = 2*A001629(n) - 3*A001629(n-1). - Johannes W. Meijer, Jun 27 2011

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

Cf. A000045, A001629.

A191830:= proc(n) option remember: if n<=1 then 0 else procname(n-1)+procname(n-2)+A000045(n-5) fi: end proc: with(combinat): A000045:=fibonacci: seq(A191830(n),n=0..30); # Johannes W. Meijer, Jun 27 2011

CoefficientList[Series[x^2(2-3x)/(1-x-x^2)^2,{x,0,40}],x] (* or *) LinearRecurrence[{2,1,-2,-1},{0,0,2,1},40] (* Harvey P. Dale, Mar 16 2015 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-2,1,2]^n*[0;0;2;1])[1,1] \\ Charles R Greathouse IV, Jul 06 2017

G.f.: x^2*(2-3*x)/(1-x-x^2)^2.
a(n) = a(n-1) + a(n-2) + A000045(n-5), a(0) = a(1) = 0.
a(0)=0, a(1)=0, a(2)=2, a(3)=1, a(n)=2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4). - Harvey P. Dale, Mar 16 2015

A245825 Triangle read by rows: T(n,k) is the number of the vertices of the Fibonacci cube G_n that have degree k (0<=k<=n).

Original entry on oeis.org

1, 0, 2, 0, 2, 1, 0, 1, 3, 1, 0, 0, 5, 2, 1, 0, 0, 3, 7, 2, 1, 0, 0, 1, 10, 7, 2, 1, 0, 0, 0, 9, 14, 8, 2, 1, 0, 0, 0, 4, 23, 16, 9, 2, 1, 0, 0, 0, 1, 22, 34, 19, 10, 2, 1, 0, 0, 0, 0, 14, 50, 44, 22, 11, 2, 1, 0, 0, 0, 0, 5, 55, 77, 56, 25, 12, 2, 1, 0, 0, 0, 0, 1, 40, 117, 106, 69, 28, 13, 2, 1, 0, 0, 0, 0, 0, 20, 131, 188, 140, 83, 31, 14, 2, 1

Offset: 0

Emeric Deutsch, Aug 03 2014

The Fibonacci cube G_n is obtained from the n-cube Q_n by removing all the vertices that contain two consecutive 1s.
Sum of entries in row n is the Fibonacci number F_{n+2}.
Sum of entries in column k (k>=1) is the Fibonacci number F_{2k+3}. - Emeric Deutsch, Jun 22 2015
Sum(k*T(n,k), k=0..n) = 2*sum(F(k)*F(n+1-k),k=0..n+1) = 2*A001629(n+1).

			Row 2 is 0,2,1 because the Fibonacci cube G_2 is the path-tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2.
Triangle starts:
1;
0,2;
0,2,1;
0,1,3,1;
0,0,5,2,1;
0,0,3,7,2,1;
0,0,1,10,7,2,1;

S. Klavzar, M. Mollard, M. Petkovsek, The degree sequence of Fibonacci and Lucas cubes, Discrete Math., 311, 2011, 1310-1322.
S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522

Cf. A000045, A001519, A001629.

T := proc (n, k) options operator, arrow: sum(binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1), i = 0 .. k) end proc: seq(seq(T(n, k), k = 0 .. n), n = 0 .. 13);

T(n,k) = sum(binomial(n-2i, k-i)*binomial(i+1,n-k-i+1), i=0..k).

G.f.: (1 + t*z + (1 - t)*t*z^2)/((1 - t*z)*(1 - t*z^2) - t*z^3).

cp's OEIS Frontend

A385817 Irregular triangle read by rows listing the lengths of maximal runs (sequences of consecutive elements increasing by 1) of binary indices, duplicate rows removed.

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A083368 a(n) is the position of the highest one in A003754(n).

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A123585 Triangle T(n,k), 0<=k<=n, given by [1, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A246177 Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) such that the area between the x-axis and the path is k.

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A332724 Number of length n - 1 ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.

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PARI

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A039913 Triangular "Fibonacci array".

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A091562 Triangle read by rows, related to Pascal's triangle, starting with 1, 0, 0.

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A128502 Convolution array for Chebyshev's S(n,x)=U(n,x/2) polynomials.

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A191830 Expansion of x^2(2-3x)/(1-x-x^2)^2.