A026150 -id:A026150 - cp's OEIS frontend

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

A162516 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 1, 3, 12, 0, 1, 6, 25, 8, 16, 1, 10, 45, 40, 80, 0, 1, 15, 75, 121, 252, 48, 64, 1, 21, 119, 287, 644, 336, 448, 0, 1, 28, 182, 588, 1457, 1360, 1888, 256, 256, 1, 36, 270, 1092, 3033, 4176, 6240, 2304, 2304, 0, 1, 45, 390, 1890, 5925, 10801, 17780, 11680, 12160, 1280, 1024

Offset: 0

Clark Kimberling, Jul 05 2009

			First six rows:
  1;
  1,  0;
  1,  1,  4;
  1,  3, 12,  0;
  1,  6, 25,  8, 16;
  1, 10, 48, 40, 80, 0;

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

Cf. A162514, A162515, A162517.
Cf. A000217, A026150, A084057, A125818, A199572.
For fixed k, the sequences P(n,k), for n=1,2,3,4,5, are A084057, A084059, A146963, A081342, A081343, respectively.

m:=12;
p:= func< n,x | ((x+Sqrt(x+4))^n + (x-Sqrt(x+4))^n)/2 >;
R:=PowerSeriesRing(Rationals(), m+1);
T:= func< n,k | Coefficient(R!( p(n,x) ), n-k) >;
[T(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Jul 09 2023

P[n_, x_]:= P[n, x]= ((x+Sqrt[x+4])^n + (x-Sqrt[x+4])^n)/2;
T[n_, k_]:= Coefficient[Series[P[n, x], {x,0,n-k+1}], x, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 08 2020; Jul 09 2023 *)

def p(n,x): return ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2
def T(n,k):
    P. = PowerSeriesRing(QQ)
    return P( p(n,x) ).list()[n-k]
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 09 2023

P(n,x) = 2*x*P(n-1,x) - (x^2 -x -4)*P(n-2,x).
From G. C. Greubel, Jul 09 2023: (Start)
T(n, k) = [x^(n-k)] ( ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2 ).
T(n, 1) = A000217(n-1), n >= 1.
T(n, n) = A199572(n).
Sum_{k=0..n} T(n, k) = A084057(n).
Sum_{k=0..n} 2^k*T(n, k) = A125818(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A026150(n).
Sum_{k=0..n} (-2)^k*T(n, k) = A133343(n). (End)

A052948 Expansion of g.f.: (1-2x)/(1-3x+2*x^3).

Original entry on oeis.org

1, 1, 3, 7, 19, 51, 139, 379, 1035, 2827, 7723, 21099, 57643, 157483, 430251, 1175467, 3211435, 8773803, 23970475, 65488555, 178918059, 488813227, 1335462571, 3648551595, 9968028331, 27233159851, 74402376363, 203271072427, 555346897579, 1517235940011, 4145165675179

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 3, s(n) = 3.

In general, a(n,m,j,k) = (2/m)*Sum_{r=1..m-1} sin(j*r*Pi/m)*sin(k*r*Pi/m)*(1+2*cos(Pi*r/m))^n is the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = j, s(n) = k. - Herbert Kociemba, Jun 02 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Denis Chebikin and Richard Ehrenborg, The f-vector of the descent polytope, arXiv:0812.1249 [math.CO], 2008-2010; Disc. Comput. Geom., 45 (2011), 410-424.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1007
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Alina F. Y. Zhao, Bijective proofs for some results on the descent polytope, Australasian Journal of Combinatorics, Volume 65(1) (2016), Pages 45-52.
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).

Cf. A026150, A077846.

a:=[1,1,3];; for n in [4..30] do a[n]:=3*a[n-1]-2*a[n-3]; od; a; # G. C. Greubel, Oct 21 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-3*x+2*x^3) )); // G. C. Greubel, Oct 21 2019

spec := [S,{S=Sequence(Prod(Union(Sequence(Prod(Sequence(Z),Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((1-2*x)/(1-3*x+2*x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 21 2019

CoefficientList[Series[(1-2x)/(1-3x+2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,0,-2},{1,1,3},30] (* Harvey P. Dale, Aug 22 2012 *)

PARI
```
Vec((1-2*x)/(1-3*x+2*x^3)+O(x^30))
```

from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,1,2,2, lambda n: -1); [next(it) for i in range(0,29)] # Zerinvary Lajos, Jul 09 2008

a(n) = 2*a(n-1) + 2*a(n-2) - 1.
a(n) = Sum_{alpha=RootOf(1-3*z+2*z^3)} alpha^(-n)/3.
a(n) = (1 + (1+sqrt(3))^n + (1-sqrt(3))^n)/3. Binomial transform of A025192 (with interpolated zeros). - Paul Barry, Sep 16 2003
a(n) = (1/3)*Sum_{k=1..5} sin(Pi*k/2)^2 * (1 + 2*cos(Pi*k/6))^n. - Herbert Kociemba, Jun 02 2004
a(0)=1, a(1)=1, a(2)=3, a(n) = 3*a(n-1) - 2*a(n-3). - Harvey P. Dale, Aug 22 2012
a(n) = A077846(n) - 2*A077846(n-1). - R. J. Mathar, Feb 27 2019
E.g.f.: exp(x)*(1 + 2*cosh(sqrt(3)*x))/3. - Stefano Spezia, Mar 02 2024

More terms from James Sellers, Jun 06 2000

Definition revised by N. J. A. Sloane, Feb 24 2011

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
  {}  {1}        {1,3}    .  .  .
      {2}        {1,4}
      {3}        {2,4}
      {4}        {1,2,4}
      {1,2}      {1,3,4}
      {2,3}
      {3,4}
      {1,2,3}
      {2,3,4}
      {1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if nT(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

A005665 Minimal number of moves for the cyclic variant of the Towers of Hanoi for 3 pegs and n disks, with the final peg one step away.

Original entry on oeis.org

0, 1, 5, 15, 43, 119, 327, 895, 2447, 6687, 18271, 49919, 136383, 372607, 1017983, 2781183, 7598335, 20759039, 56714751, 154947583, 423324671, 1156544511, 3159738367, 8632565759, 23584608255, 64434348031, 176037912575, 480944521215, 1313964867583, 3589818777599, 9807567290367

Offset: 0

N. J. A. Sloane

Original name was: Tower of Hanoi with 3 pegs and cyclic moves only (clockwise). - Jianing Song, Nov 01 2024

This looks like sequence A(0,1;2,2;3) of the family of sequences [a,b:c,d:k] considered by Gary Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Alejandre, Legend of Towers of Hanoi
J.-P. Allouche, Note on the cyclic towers of Hanoi, Theoret. Comput. Sci., 123 (1994), 3-7.
M. D. Atkinson, The Cyclic Towers of Hanoi, Info. Proc. Letters, 13 (1981), 118-119.
R. K. Guy, Letter to N. J. A. Sloane, 1976
A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 249. Book's website
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
D. G. Poole, The towers and triangles of Professor Claus (or, Pascal knows Hanoi), Math. Mag., 67 (1994), 323-344.
Index entries for linear recurrences with constant coefficients, signature (3,0,-2).
Index entries for sequences related to Towers of Hanoi

Cf. A005666, A007664, A007665, A026150 (first differences).

Cf. A338024, A292764, A338089 (4 pegs).

a005665 n = a005665_list !! (n-1)
a005665_list = 0 : 1 : 5 : zipWith (-)
               (map (* 3) $ drop 2 a005665_list) (map (* 2) a005665_list)
-- Reinhard Zumkeller, May 01 2013

[Floor(((Sqrt(3)+1)^(n+1)+(Sqrt(3)-1)^(n+1)*(-1)^n)*Sqrt(3)/6-1): n in [0..30] ]; // Vincenzo Librandi, Aug 19 2011

a[n_] := Simplify[ ((1 + Sqrt[3])^(n+1) - (1 - Sqrt[3])^(n+1))*Sqrt[3]/6 - 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 14 2011, after Paul Barry *)
LinearRecurrence[{3,0,-2},{0,1,5},40] (* Harvey P. Dale, Mar 30 2015 *)

a(n)=([0,1,0; 0,0,1; -2,0,3]^n*[0;1;5])[1,1] \\ Charles R Greathouse IV, Jun 15 2015

G.f.: x*(1+2*x)/((1-x)*(1-2*x-2*x^2)). - Simon Plouffe in his 1992 dissertation
From Paul Barry, Sep 05 2006: (Start)
a(n) = ((sqrt(3)+1)^(n+1) + (sqrt(3)-1)^(n+1)*(-1)^n)*sqrt(3)/6 - 1. (End)
a(n) = 2*a(n-1) + 2*a(n-2) + 3. - John W. Layman
a(n) = (1/(2*s3))*((1+s3)^(n+1) - (1-s3)^(n+1)) - 1 where s3 = sqrt(3).
a(n) = 3*a(n-1) - 2*a(n-3), a(0)=0, a(1)=1, a(2)=5 (from the given o.g.f.). Observed by Gary Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010
a(n) = 2*A005666(n-1) + 1. - Michel Marcus, Nov 02 2012
a(n) = Sum_{k=1..n} A026150(k). - Ivan N. Ianakiev, Nov 22 2019
E.g.f.: (1/3)*exp(x)*(-3 + 3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Nov 22 2019

More terms from Vincenzo Librandi, Aug 19 2011
Name clarified by Paul Zimmermann, Feb 21 2018
New name based on the name of A338024, A292764, and A338089 by Jianing Song, Nov 01 2024

A173299 Numerators of fractions x^n + y^n, where x + y = 1 and x^2 + y^2 = 2.

Original entry on oeis.org

1, 2, 5, 7, 19, 13, 71, 97, 265, 181, 989, 1351, 3691, 2521, 13775, 18817, 51409, 35113, 191861, 262087, 716035, 489061, 2672279, 3650401, 9973081, 6811741, 37220045, 50843527, 138907099, 94875313, 518408351, 708158977, 1934726305, 1321442641

Offset: 1

J. Lowell, Feb 15 2010

x and y are given by -A152422 and 1-A152422. - R. J. Mathar, Mar 01 2010

Letting f(n) = x^n + y^n, recurrence relation f(n) = f(n - 1) + f(n - 2)/2 implies a(n) / A173300(n) = A026150(n) / 2^(n - 1). - Nick Hobson, Jan 30 2024

			a(3) = 5 because x^3 + y^3 is 2.5 and 2.5 is 5/2.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A173300 (denominators).

Z:=PolynomialRing(Integers()); N:=NumberField(2*x^2-2*x-1); S:=[ r^n+(1-r)^n: n in [1..34] ]; [ Numerator(RationalField()!S[j]): j in [1..#S] ]; // Klaus Brockhaus, Mar 02 2010

A173299 := proc(n) local x,y ; x := (1+sqrt(3))/2 ; y := (1-sqrt(3))/2 ; expand(x^n+y^n) ; numer(%) ; end proc: # R. J. Mathar, Mar 01 2010

Module[{x=(1-Sqrt[3])/2,y},y=1-x;Table[x^n+y^n,{n,40}]]//Simplify// Numerator (* Harvey P. Dale, Aug 24 2019 *)

{ a(n) = numerator( 2 * polcoeff( lift( Mod((1+x)/2,x^2-3)^n ), 0) ) }

from fractions import Fraction
def a173299_gen(a, b):
    while True:
        yield a.numerator
        b, a = b + Fraction(a, 2), b
g = a173299_gen(1, 2)
print([next(g) for  in range(34)])  # _Nick Hobson, Feb 20 2024

a(n) = numerator of ((1 + sqrt(3))/2)^n + ((1 - sqrt(3))/2)^n.

Formula, more terms, and PARI script from Max Alekseyev, Feb 24 2010

More terms from Klaus Brockhaus and R. J. Mathar, Mar 01 2010

A084097 Square array whose rows have e.g.f. exp(x)cosh(sqrt(k)x), k>=0, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 7, 8, 1, 1, 1, 5, 10, 17, 16, 1, 1, 1, 6, 13, 28, 41, 32, 1, 1, 1, 7, 16, 41, 76, 99, 64, 1, 1, 1, 8, 19, 56, 121, 208, 239, 128, 1, 1, 1, 9, 22, 73, 176, 365, 568, 577, 256, 1, 1, 1, 10, 25, 92, 241, 576, 1093, 1552, 1393, 512, 1

Offset: 0

Paul Barry, May 11 2003

Rows are the binomial transforms of expansions of cosh(sqrt(k)*x), k >= 0.

			Array, A(n,k), begins:
.n\k.........0..1...2...3....4.....5......6......7.......8........9.......10
.0: A000012..1..1...1...1....1.....1......1......1.......1........1........1
.1: A000079..1..1...2...4....8....16.....32.....64.....128......256......512
.2: A001333..1..1...3...7...17....41.....99....239.....577.....1393.....3363
.3: A026150..1..1...4..10...28....76....208....568....1552.....4240....11584
.4: A046717..1..1...5..13...41...121....365...1093....3281.....9841....29525
.5: A084057..1..1...6..16...56...176....576...1856....6016....19456....62976
.6: A002533..1..1...7..19...73...241....847...2899...10033....34561...119287
.7: A083098..1..1...8..22...92...316...1184...4264...15632....56848...207488
.8: A084058..1..1...9..25..113...401...1593...5993...23137....88225...338409
.9: A003665..1..1..10..28..136...496...2080...8128...32896...130816...524800
10: A002535..1..1..11..31..161...601...2651..10711...45281...186961...781451
11: A133294..1..1..12..34..188...716...3312..13784...60688...259216..1125312
12: A090042..1..1..13..37..217...841...4069..17389...79537...350353..1575613
13: A125816..1..1..14..40..248...976...4928..21568..102272...463360..2153984
14: A133343..1..1..15..43..281..1121...5895..26363..129361...601441..2884575
15: A133345..1..1..16..46..316..1276...6976..31816..161296...768016..3794176
16: A120612..1..1..17..49..353..1441...8177..37969..198593...966721..4912337
17: A133356..1..1..18..52..392..1616...9504..44864..241792..1201408..6271488
18: A125818..1..1..19..55..433..1801..10963..52543..291457..1476145..7907059
25: A083578
- _Robert G. Wilson v_, Jan 02 2013
Antidiagonal triangle, T(n,k), begins:
  1;
  1,  1;
  1,  1,  1;
  1,  1,  2,  1;
  1,  1,  3,  4,  1;
  1,  1,  4,  7,  8,   1;
  1,  1,  5, 10, 17,  16,   1;
  1,  1,  6, 13, 28,  41,  32,    1;
  1,  1,  7, 16, 41,  76,  99,   64,    1;
  1,  1,  8, 19, 56, 121, 208,  239,  128,    1;
  1,  1,  9, 22, 73, 176, 365,  568,  577,  256,   1;
  1,  1, 10, 25, 92, 241, 576, 1093, 1552, 1393, 512,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf  A140895, A221131.
Rows: A000012, A000079, A001333, A026150, A046717, A084057, A002533, A083098, A084058, A003665,
Rows: A002535, A133294, A090042, A125816, A133343, A133345, A120612, A133356, A125818, A083578.
Columns: A000012, A000012, A000027, A016777, A028884, A134593.
Rows include A011782, A001333, A026150, A046717, A002533.

function A084097(n,k)
  if k eq 0 then return 1;
  else return k*2^(k-1)*(&+[ Binomial(k-j,j)*((n-k-1)/4)^j/(k-j): j in [0..Floor(k/2)]]);
  end if; return A084097; end function;
[A084097(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 15 2022

T[j_, k_] := Expand[((1 + Sqrt[j])^k + (1 - Sqrt[j])^k)/2]; T[1, 0] = 1; Table[ T[j - k, k], {j, 0, 11}, {k, 0, j}] // Flatten (* Robert G. Wilson v, Jan 02 2013 *)

def A084097(n,k):
    if (k==0): return 1
    else: return k*2^(k-1)*sum( binomial(k-j,j)*((n-k-1)/4)^j/(k-j) for j in range( (k+2)//2 ) )
flatten([[A084097(n,k) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Oct 15 2022

From Robert G. Wilson v, Jan 02 2013: (Start)
A(n, k) = (1/2)*( (1 + sqrt(n))^k + (1 - sqrt(n))^k ) (array).
T(n, k) = A(n-k, k). (End)
T(n, k) = Sum_{j=0..floor(k/2)} binomial(k-j, j)*((n-k-1)/4)^j/(k-j), with T(n, 0) = 1 (antidiagonal triangle T(n,k)). - G. C. Greubel, Oct 15 2022

Edited by N. J. A. Sloane, Jul 14 2010

A122016 Riordan array(1, x(1+2x)/(1-x)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 3, 6, 1, 0, 3, 15, 9, 1, 0, 3, 24, 36, 12, 1, 0, 3, 33, 90, 66, 15, 1, 0, 3, 42, 171, 228, 105, 18, 1, 0, 3, 51, 279, 579, 465, 153, 21, 1, 0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1, 0, 3, 69, 576, 2172, 3858, 3258, 1344, 276, 27, 1

Offset: 0

Philippe Deléham, Sep 24 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Rising and falling diagonals are A078010 and A122552.

			Triangle begins:
  1;
  0, 1;
  0, 3,  1;
  0, 3,  6,   1;
  0, 3, 15,   9,    1;
  0, 3, 24,  36,   12,    1;
  0, 3, 33,  90,   66,   15,   1;
  0, 3, 42, 171,  228,  105,  18,   1;
  0, 3, 51, 279,  579,  465, 153,  21,  1;
  0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1;

Huyile Liang, Jinyang Zhang, and Yu Wang, Some properties of the matrix related to q-coloured coordination number, Filomat (2024) Vol. 38, No. 4, 1465-1477. See p. 1466.

Cf. Diagonals A000012, A008585, A062741, columns A000007, A122553, A122709, row sums A026150.

Cf. A078010, A084938, A102900, A117575, A122117, A122552, A147518.

T[n_,k_]:=SeriesCoefficient[(1-x)/(1-(y+1)*x-2*y*x^2),{x,0,n},{y,0,k}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Dec 27 2023 *)

Sum_{k=0..n} T(n,k)*x^(n-k) = A026150(n), A102900(n) for x = 1, 2.
T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-1). - Philippe Deléham, Sep 25 2006
G.f.: (1-x)/(1-(y+1)*x-2*y*x^2). - Philippe Deléham, Jan 31 2012
Sum_{k=0..n} T(n,k)*x^k = A117575(n+1), A000007(n), A026150(n), A122117(n), A147518(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 31 2012

More terms from Stefano Spezia, Dec 27 2023

A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1

Offset: 0

Charles L. Hohn, May 31 2011

			1, 0,  0,   0,    0,    0,     0,      0,       0,        0,        0, ...
1, 1,  2,   4,    8,   16,    32,     64,     128,      256,      512, ...
1, 1,  3,   7,   17,   41,    99,    239,     577,     1393,     3363, ...
1, 1,  4,  10,   28,   76,   208,    568,    1552,     4240,    11584, ...
1, 2,  8,  32,  128,  512,  2048,   8192,   32768,   131072,   524288, ...
1, 2,  9,  38,  161,  682,  2889,  12238,   51841,   219602,   930249, ...
1, 2, 10,  44,  196,  872,  3880,  17264,   76816,   341792,  1520800, ...
1, 2, 11,  50,  233, 1082,  5027,  23354,  108497,   504050,  2341691, ...
1, 2, 12,  56,  272, 1312,  6336,  30592,  147712,   713216,  3443712, ...
1, 3, 18, 108,  648, 3888, 23328, 139968,  839808,  5038848, 30233088, ...
1, 3, 19, 117,  721, 4443, 27379, 168717, 1039681,  6406803, 39480499, ...
1, 3, 20, 126,  796, 5028, 31760, 200616, 1267216,  8004528, 50561600, ...
1, 3, 21, 135,  873, 5643, 36477, 235791, 1524177,  9852435, 63687141, ...
1, 3, 22, 144,  952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
...

Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.
Cf. A191348 which uses ceiling() in place of floor().

T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
matrix(9,9, n, k, T(n-1,k-1)) \\ Michel Marcus, Aug 22 2019

T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 22 2019

For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019

T(n, k) = Sum_{i=0..floor((k+1)/2)} binomial(k, 2*i)*floor(sqrt(n))^(k-2*i)*n^i for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019

Previous Showing 31-40 of 53 results. Next

cp's OEIS Frontend

A184686 T(n,k)=Number of (n+1)X(k+1) binary arrays with every 2X2 subblock singular.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A201701 Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A162516 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A052948 Expansion of g.f.: (1-2*x)/(1-3*x+2*x^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A005665 Minimal number of moves for the cyclic variant of the Towers of Hanoi for 3 pegs and n disks, with the final peg one step away.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

Extensions

A173299 Numerators of fractions x^n + y^n, where x + y = 1 and x^2 + y^2 = 2.

Views

Author

Keywords

Comments

Examples

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

A052948 Expansion of g.f.: (1-2x)/(1-3x+2*x^3).

A084097 Square array whose rows have e.g.f. exp(x)cosh(sqrt(k)x), k>=0, read by ascending antidiagonals.

A122016 Riordan array(1, x(1+2x)/(1-x)).