A005314 -id:A005314 - cp's OEIS frontend

A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 2, 0, 3, 1, 0, 4, 4, 0, 0, 5, 10, 1, 0, 0, 6, 20, 6, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (x/(1-x)^2, x^2/(1-x)^2).
Mirror image of triangle in A119900.
A203322*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 05 2011
From Gus Wiseman, Jul 07 2025: (Start)
Also the number of subsets of {1..n} containing n with k maximal runs (sequences of consecutive elements increasing by 1). For example, row n = 5 counts the following subsets:
  {5}          {1,5}      {1,3,5}
  {4,5}        {2,5}
  {3,4,5}      {3,5}
  {2,3,4,5}    {1,2,5}
  {1,2,3,4,5}  {1,4,5}
               {2,3,5}
               {2,4,5}
               {1,2,3,5}
               {1,2,4,5}
               {1,3,4,5}
For anti-runs instead of runs we have A053538.
Without requiring n see A210039, A202023, reverse A098158, A109446.
(End)

			Triangle begins :
1
2, 0
3, 1, 0
4, 4, 0, 0
5, 10, 1, 0, 0
6, 20, 6, 0, 0, 0
7, 35, 21, 1, 0, 0, 0
8, 56, 56, 8, 0, 0, 0, 0

Cf. A007318, A005314 (antidiagonal sums), A119900, A084938, A130595, A203322.
Column k = 1 is A000027.
Row sums are A000079.
Column k = 2 is A000292.
Without zeros we have A034867.
Last nonzero term in each row appears to be A124625.
A034839 counts subsets by number of maximal runs, for anti-runs A384893.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
Cf. A000045, A000071, A001629, A010027, A053538, A208342, A210034, A245563, A268193, A384177, A384890.
Cf. A098158, A109446, A202023, A210039.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Length[Split[#,#2==#1+1&]]==k&]],{n,12},{k,n}] (* Gus Wiseman, Jul 07 2025 *)

G.f.: 1/((1-x)^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13 respectively.
T(n,k) = binomial(n+1,2k+1).
T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 15 2012

A077954 Expansion of 1/(1-x+2*x^2-x^3) in powers of x.

Original entry on oeis.org

1, 1, -1, -2, 1, 4, 0, -7, -3, 11, 10, -15, -24, 16, 49, -7, -89, -26, 145, 108, -208, -279, 245, 595, -174, -1119, -176, 1888, 1121, -2831, -3185, 3598, 7137, -3244, -13920, -295, 24301, 10971, -37926, -35567, 51256, 84464, -53615, -171287, 20407, 309366, 97265, -501060, -386224, 713161

Offset: 0

N. J. A. Sloane, Nov 17 2002

			G.f. = 1 + x - x^2 - 2*x^3 + x^4 + 4*x^5 - 7*x^7 - 3*x^8 + 11*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (1,-2,1).

Cf. A000931, A005314.

a:=[1,1,-1];; for n in [4..50] do a[n]:=a[n-1]-2*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Aug 07 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x+2*x^2-x^3) )); // G. C. Greubel, Aug 07 2019

seq(coeff(series(1/(1-x+2*x^2-x^3), x, n+1), x, n), n = 0..50); # G. C. Greubel, Aug 07 2019

a[ n_] := If[ n < 0, SeriesCoefficient[ x^3 / (1 - 2 x + x^2 - x^3), {x, 0, -n}], SeriesCoefficient[ 1 / (1 - x + 2 x^2 - x^3), {x, 0, n}]]
LinearRecurrence[{1,-2,1}, {1,1,-1}, 50] (* G. C. Greubel, Aug 07 2019 *)

{a(n) = if( n<0,  polcoeff( x^3 / (1 - 2*x + x^2 - x^3) + x * O(x^-n), -n), polcoeff( 1 / (1 - x + 2*x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, Sep 18 2012 */

(1/(1-x+2*x^2-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Aug 07 2019

a(0)=1, a(1)=1, a(2)=-1, a(n) = a(n-1) -2*a(n-2) +a(n-3) for n>=3. - Philippe Deléham, Sep 15 2006
a(n) = A000931(-2*n). - Michael Somos, Sep 18 2012
a(n) = A005314(-n-2). - Michael Somos, Dec 13 2013
a(n) = a(n-1) - 2*a(n-2) + a(n-3) for all n in Z. - Michael Somos, Dec 13 2013

A136444 a(n) = Sum_{k=0..n} kbinomial(n-k, 2k).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 12, 25, 51, 101, 197, 381, 731, 1392, 2634, 4958, 9290, 17337, 32239, 59760, 110460, 203651, 374593, 687567, 1259597, 2303449, 4205493, 7666560, 13956532, 25374108, 46076436, 83575025, 151431099, 274108826, 495708364, 895670733, 1617003823, 2916984121

Offset: 0

Don Knuth, Apr 04 2008

Consider four related sequences: A_n = sum C(n-k, 2*k), B_n = sum C(n-k, 2*k+1), A^*_n = sum k*C(n-k, 2*k), B^*_n = sum k*C(n-k, 2*k+1).
Sequence A_n, with generating function (1-z)/p(z) where p(z) = 1 - 2*z + z^2 - z^3, is A005251.
Sequence B_n, with generating function z/p(z), is A005314.
Sequence A^*_n is the present sequence.
Sequence B^*_n is A118430, but shifted one place so that the generating function is z^4/p(z)^2 instead of z^3/p(z)^2.
These sequences have many interrelations; for example,
B_{n+1} - B_n = A_n; B^*_{n+1} - B^*_n = A^*_n;
A_{n+1} - A_n = B_{n-1}; A^*{n+1} - A^*_n = B^*{n-1} + B_{n-1}.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), #10.6.8, partitions of [n] with 2 blocks with 1 peak.

Cf. A005251, A005314, A118430, A136445, A137356-A137361.

[&+[k*Binomial(n-k, 2*k): k in [0..n]]: n in [0..40]]; // Bruno Berselli, Feb 13 2015

a:= n-> (Matrix([[0,0,1,1,-3,-5]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [4,-6,6,-5,2,-1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..37);  # Alois P. Heinz, Aug 13 2008

a[n_] := ({0, 0, 1, 1, -3, -5} . MatrixPower[ Table[If[i == j-1, 1, If[j == 1, {4, -6, 6, -5, 2, -1}[[i]], 0]], {i, 6}, {j, 6}], n])[[1]]; Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)
CoefficientList[Series[x^3 (1 - x)/(1 - 2 x + x^2 - x^3)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 15 2015 *)

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

a(n) ~ c * d^n * n, where d = A109134 = 1.75487766624669276... is the root of the equation d*(d-1)^2 = 1, c = 0.072838349685011... is the root of the equation 529*c^3 - 207*c^2 + 26*c = 1. - Vaclav Kotesovec, May 25 2015

A077930 Expansion of (1-x)^(-1)/(1+2*x+x^2+x^3).

Original entry on oeis.org

1, -1, 2, -3, 6, -10, 18, -31, 55, -96, 169, -296, 520, -912, 1601, -2809, 4930, -8651, 15182, -26642, 46754, -82047, 143983, -252672, 443409, -778128, 1365520, -2396320, 4205249, -7379697, 12950466, -22726483, 39882198, -69988378, 122821042, -215535903, 378239143, -663763424

Offset: 0

N. J. A. Sloane, Nov 17 2002

Signed version of A060945: a(n) = (-1)^n * A060945(n).

Index entries for linear recurrences with constant coefficients, signature (-1,1,0,1).

All of A060945, A077930, A181532 are variations of the same sequence. - N. J. A. Sloane, Mar 04 2012

CoefficientList[Series[1/(1-x)/(1+2x+x^2+x^3),{x,0,50}],x] (* or *) LinearRecurrence[ {-1,1,0,1},{1,-1,2,-3},50] (* Harvey P. Dale, Feb 20 2013 *)

Vec((1-x)^(-1)/(1+2*x+x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

a(n+1)-a(n) = (-1)^(n+1)*A005314(n+2). - R. J. Mathar, Mar 14 2011

a(0)=1, a(1)=-1, a(2)=2, a(3)=-3, a(n)=-a(n-1)+a(n-2)+a(n-4). - Harvey P. Dale, Feb 20 2013

A099529 Expansion of (1+x)^2/((1+x)^2+x^3).

Original entry on oeis.org

1, 0, 0, -1, 2, -3, 5, -9, 16, -28, 49, -86, 151, -265, 465, -816, 1432, -2513, 4410, -7739, 13581, -23833, 41824, -73396, 128801, -226030, 396655, -696081, 1221537, -2143648, 3761840, -6601569, 11584946, -20330163, 35676949, -62608681, 109870576, -192809420, 338356945, -593775046, 1042002567

Offset: 0

Paul Barry, Oct 20 2004

Binomial transform has g.f. 1/(1-x+x^3) (A050935(n+2)).

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

a(n)=-2a(n-1)-a(n-2)-a(n-3); a(n)=sum{j=0..n, sum{k=0..floor(j/3), C(n, j)(-1)^(n-j)C(j-2k, k)(-1)^k}}.

a(n)=(-1)^n*A005314(n-2). [From R. J. Mathar, Nov 26 2008]

A259967 a(n) = a(n-1) + a(n-2) + a(n-4).

Original entry on oeis.org

3, 2, 2, 5, 10, 17, 29, 51, 90, 158, 277, 486, 853, 1497, 2627, 4610, 8090, 14197, 24914, 43721, 76725, 134643, 236282, 414646, 727653, 1276942, 2240877, 3932465, 6900995, 12110402, 21252274, 37295141, 65448410, 114853953, 201554637, 353703731, 620706778

Offset: 0

N. J. A. Sloane, Jul 11 2015

Also the number of maximal independent vertex sets (and minimal vertex covers) in the n-gear graph. - Eric W. Weisstein, May 25 2017

Also the number of chordless cycles in the n-antiprism graph for n >= 4. - Eric W. Weisstein, Jan 02 2018

R. K. Guy, Letter to N. J. A. Sloane, Feb 05 1986.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Nazim Fatès, Biswanath Sethi, Sukanta Das, On the reversibility of ECAs with fully asynchronous updating: the recurrence point of view, Preprint, hal-01571847, 2017.
R. K. Guy, Letter to N. J. A. Sloane, Feb 1986
Bojan Vučković and Miodrag Živković, Row Space Cardinalities Above 2^(n - 2) + 2^(n - 3), ResearchGate, January 2017, p. 3.
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Chordless Cycle
Eric Weisstein's World of Mathematics, Gear Graph
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Minimal Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Cf. A109377, A259968, A259969.

a259967 n = a259967_list !! n
a259967_list = 3 : 2 : 2 : 5 : zipWith3 (((+) .) . (+))
   a259967_list (drop 2 a259967_list) (drop 3 a259967_list)
-- Reinhard Zumkeller, Jul 12 2015

I:=[3,2,2,5]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Sep 26 2017

f:= gfun:-rectoproc({-a(n+3)+2*a(n+2)-a(n+1)+a(n), a(0) = 3, a(1) = 2, a(2) = 2},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jul 18 2016

Abs @ CoefficientList[Series[(x - 1) (x - 3)/(-1 + 2 x - x^2 + x^3), {x, 0, 36}], x] (* Michael De Vlieger, Jul 18 2016 *)
LinearRecurrence[{2, -1, 1}, {2, 2, 5}, 20] (* Eric W. Weisstein, May 25 2017 *)
Table[RootSum[-1 + # - 2 #^2 + #^3 &, #^n &], {n, 20}] (* Eric W. Weisstein, May 25 2017 *)
RootSum[-1 + # - 2 #^2 + #^3 &, #^Range[0, 20] &] (* Eric W. Weisstein, Jan 02 2018 *)

x='x+O('x^50); Vec((x-1)*(x-3)/(1-2*x+x^2-x^3)) \\ G. C. Greubel, May 24 2017

G.f.: (x-1)*(x-3) / (1 -2*x +x^2 -x^3). - R. J. Mathar, Jul 15 2015
a(n) = -4*A005314(n) +3*A005314(n+1) +A005314(n-1). - R. J. Mathar, Jul 15 2015
a(n) = Sum_{i=1..3} r_i^n where r_i are the roots of x^3-2*x^2+x-1. - Robert Israel, Jul 18 2016
a(n) = A109377(n-2) for n > 1. - Georg Fischer, Oct 09 2018

A224838 Triangle read by rows, obtained from triangle A011973 by reading that array from right to left along the irregular paths shown in the figure.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 4, 6, 5, 1, 1, 10, 10, 6, 1, 1, 5, 20, 15, 7, 1, 6, 15, 35, 21, 8, 1, 1, 21, 35, 56, 28, 9, 1, 1, 7, 56, 70, 84, 36, 10, 1, 8, 28, 126, 126, 120, 45, 11, 1, 1, 36, 84, 252, 210, 165, 55, 12, 1, 1, 9, 120, 210, 462, 330, 220, 66, 13, 1

Offset: 1

John Molokach, Jul 21 2013

The successive rows have lengths 1,2,2; 3,4,4; 5,6,6; 7,8,8; ...
Sum of row n is A005314(n).
Old definition was: "Triangle of falling diagonals of A011973 (with rows displayed as centered text)."

			First 11 rows of the triangle:
  1;
  1,  1;
  2,  1;
  1,  3,  1;
  1,  3,  4,  1;
  4,  6,  5,  1;
  1, 10, 10,  6,  1;
  1,  5, 20, 15,  7,  1;
  6, 15, 35, 21,  8,  1;
  1, 21, 35, 56, 28,  9,  1;
  1,  7, 56, 70, 84, 36, 10,  1;

N. J. A. Sloane, Construction of present triangle by reading triangle A011973 from right to left along the paths indicated.

Cf. A005314, A227300, A001973, A000045, A004396.

Table[Reverse[Table[Binomial[n - Floor[(k + 1)/2], n - Floor[(3 k - 1)/2]], {k, Floor[(2 n + 2)/3]}]], {n, 13}] (* T. D. Noe, Jul 25 2013 *)
Column[Table[Binomial[n - Floor[(4 n + 15 - 6 k + (-1)^k)/12], n - Floor[(4 n + 15 - 6 k + (-1)^k)/12] - Floor[(2 n - 1)/3] + k - 1], {n, 1, 25}, {k, 1, Floor[(2 n + 2)/3]}]] (* John Molokach, Jul 25 2013 *)

r(n) = binomial(n-floor((4n+15-6k+(-1)^k)/12), n-floor((4n+15-6k+(-1)^k)/12)-floor((2n-1)/3)+k-1), k = 1..floor((2n+2)/3).

R(n) = binomial(n-floor((k+1)/2), n-floor((3k-1)/2)), k = 1..floor((2n+2)/3), gives the terms of each row in reverse order.

Entry revised by N. J. A. Sloane, Jul 07 2024

A289692 The number of partitions of [n] with exactly 2 blocks without peaks.

Original entry on oeis.org

0, 1, 2, 4, 8, 15, 27, 48, 85, 150, 264, 464, 815, 1431, 2512, 4409, 7738, 13580, 23832, 41823, 73395, 128800, 226029, 396654, 696080, 1221536, 2143647, 3761839, 6601568, 11584945

Offset: 1

R. J. Mathar, Jul 09 2017

Muniru A Asiru, Table of n, a(n) for n = 1..300
T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Table 1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-1).

Cf. A005251 (first differences), A289693 (3 blocks), A289694 (4 blocks).

a:=[0, 1, 2, 4]; for n in [5..10^2] do a[n]:=3*a[n-1]-3*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Jan 25 2018

I:=[0, 1, 2, 4]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jan 26 2018

a := proc(n) option remember: if n = 1 then 0 elif n = 2 then 1 elif n=3 then 2 elif n=4 then 4 elif  n >= 5 then 3*procname(n-1) -3*procname(n-2)+2*procname(n-3)-procname(n-4) fi; end:
seq(a(n), n = 0..100); # Muniru A Asiru, Jan 25 2018

LinearRecurrence[{3, -3, 2, -1}, {0, 1, 2, 4}, 40] (* Vincenzo Librandi, Jan 26 2018 *)

From Colin Barker, Nov 07 2017: (Start)
G.f.: x^2*(1 - x + x^2) / ((1 - x)*(1 - 2*x + x^2 - x^3)).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) for n>4. (End)
a(n) = A077855(n-2) - A005314(n-2) for n>1. - John Molokach, Jan 23 2018
a(n) - a(n-1) = A005251(n). - R. J. Mathar, Mar 11 2021

A193736 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (n+1)-st Fibonacci polynomial and q(n,x) = (x+1)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 8, 8, 3, 1, 5, 13, 19, 15, 5, 1, 6, 19, 36, 42, 28, 8, 1, 7, 26, 60, 91, 89, 51, 13, 1, 8, 34, 92, 170, 216, 182, 92, 21, 1, 9, 43, 133, 288, 446, 489, 363, 164, 34, 1, 10, 53, 184, 455, 826, 1105, 1068, 709, 290, 55, 1, 11, 64, 246, 682, 1414, 2219, 2619, 2266, 1362, 509, 89

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

			First six rows:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  4,  2;
  1,  4,  8,  8,  3;
  1,  5, 13, 19, 15,  5;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A005314 (diagonal sums), A052542 (row sums), A077962.

Cf. A029907, A193722, A193737, A324969.

function T(n,k) // T = A193736
  if k lt 0 or n lt 0 then return 0;
  elif n lt 3 then return Binomial(n,k);
  else return T(n-1, k) + T(n-1, k-1) + T(n-2, k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023

(* First program *)
p[0, x_] := 1
p[n_, x_] := Fibonacci[n + 1, x] /; n > 0
q[n_, x_] := (x + 1)^n
t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
t[n_, n_] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n - k + 1, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193736 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]          (* A193737 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<3, Binomial[n, k], T[n-1,k] +T[n-1,k-1] +T[n -2,k-2]];
Table[T[n, k], {n,0,12}, {k,0,n}]//TableForm (* G. C. Greubel, Oct 24 2023 *)

def T(n,k): # T = A193736
    if (n<3): return binomial(n,k)
    else: return T(n-1,k) +T(n-1,k-1) +T(n-2,k-2)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,2) = 1; T(n,k) = 0 if k<0 or k>n; T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2). - Philippe Deléham, Feb 13 2020
From G. C. Greubel, Oct 24 2023: (Start)
T(n, k) = A193737(n, n-k).
T(n, n) = Fibonacci(n) + [n=0] = A324969(n+1).
T(n, n-1) = A029907(n).
Sum_{k=0..n} T(n, k) = A052542(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A005314(n) + [n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = [n=0] + A077962(n-1). (End)

A099557 Slanted Pascal's triangle, read by rows, such that T(n,k) = binomial(n-[k/2],k) for [n*2/3]>=k>=0, where [x]=floor(x).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 3, 1, 0, 1, 5, 6, 4, 0, 0, 1, 6, 10, 10, 1, 0, 0, 1, 7, 15, 20, 5, 1, 0, 0, 1, 8, 21, 35, 15, 6, 0, 0, 0, 1, 9, 28, 56, 35, 21, 1, 0, 0, 0, 1, 10, 36, 84, 70, 56, 7, 1, 0, 0, 0, 1, 11, 45, 120, 126, 126, 28, 8, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Oct 22 2004

Row sums form A005314. Antidiagonal sums form A099558.

			Rows begin:
[1],
[1,1],
[1,2,0],
[1,3,1,0],
[1,4,3,1,0],
[1,5,6,4,0,0],
[1,6,10,10,1,0,0],
[1,7,15,20,5,1,0,0],
[1,8,21,35,15,6,0,0,0],
[1,9,28,56,35,21,1,0,0,0],
[1,10,36,84,70,56,7,1,0,0,0],...
and can be derived from Pascal's triangle
by shifting each column k down by [k/2] rows.

Cf. A005314, A099558.

{T(n,k)=polcoeff(polcoeff((1-x+x*y)/((1-x)^2-x^3*y^2)+x*O(x^n),n,x)+y*O(y^k),k,y)}

G.f.: (1-x+x*y)/((1-x)^2-x^3*y^2).

cp's OEIS Frontend

A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A259967 a(n) = a(n-1) + a(n-2) + a(n-4).

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A224838 Triangle read by rows, obtained from triangle A011973 by reading that array from right to left along the irregular paths shown in the figure.

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A136444 a(n) = Sum_{k=0..n} kbinomial(n-k, 2k).