A056241 -id:A056241 - cp's OEIS frontend

A104027 Triangle, read by rows, equal to the matrix inverse of A056241, which is formed from the even-indexed trinomial coefficients.

1, -1, 1, 2, -3, 1, -7, 12, -6, 1, 41, -73, 41, -10, 1, -376, 675, -390, 105, -15, 1, 5033, -9048, 5256, -1446, 225, -21, 1, -92821, 166901, -97034, 26796, -4242, 427, -28, 1, 2257166, -4058703, 2359939, -652054, 103515, -10570, 742, -36, 1, -69981919, 125837748, -73169550, 20218251, -3210939

Offset: 0

Paul D. Hanna, Feb 26 2005

Column 0 forms signed Hammersley's polynomial p_n(1) (A006846). Column 1 forms A104028.

Triangle T(n,k), 0<=k<=n, read by rows, given by [ -1, -1, -3, -4, -7, -9, -13, -16, -21, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938; see A004652 : 0, 1, 1, 3, 4, 7, 9, 13, ... - Philippe Deléham, Sep 26 2005

			Rows begin:
1;
-1,1;
2,-3,1;
-7,12,-6,1;
41,-73,41,-10,1;
-376,675,-390,105,-15,1;
5033,-9048,5256,-1446,225,-21,1;
-92821,166901,-97034,26796,-4242,427,-28,1;
2257166,-4058703,2359939,-652054,103515,-10570,742,-36,1; ...

Cf. A056241, A104028, A027907.

PARI
```
T(n,k)=if(n
				
```

A104028 Column 1 of triangle A104027, which is the matrix inverse of the triangle A056241 of even-indexed trinomial coefficients.

Original entry on oeis.org

1, -3, 12, -73, 675, -9048, 166901, -4058703, 125837748, -4845013765, 226796981895, -12684595018992, 835391818484873, -63990023222817531, 5640684058036591260, -566948619030797914657, 64452061572236327204235, -8228252550026752605862344

Offset: 0

Paul D. Hanna, Feb 26 2005

sign

Column 0 of triangle A104027 forms signed Hammersley's polynomial p_n(1) (A006846).

Cf. A104027, A006846.

{a(n)=if(n<0,0,((matrix(n+2,n+2,m,j, if(m>=j,polcoeff((1+x+x^2)^(m-1)+O(x^(2*j)),2*j-2))))^-1)[n+2,2])}

A124302 Number of set partitions with at most 3 blocks; number of Dyck paths of height at most 4; dimension of space of symmetric polynomials in 3 noncommuting variables.

Original entry on oeis.org

1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481

Offset: 0

Mike Zabrocki, Oct 25 2006

Row sums of triangle in A056241. - Philippe Deléham, Oct 30 2006
Row sums of triangle in A147746. - Philippe Deléham, Dec 04 2008
Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008
Number of nonisomorphic graded posets with 0 and 1 and uniform Hasse graph of rank n with no 3-element antichain. (Uniform used in the sense of Retakh, Serconek and Wilson.  Graded used in Stanley's sense that every maximal chain has the same length n.) - David Nacin, Feb 26 2012
Number of Dyck paths of length 2n and height at most 4. - Ira M. Gessel, Aug 06 2012

			There are 15 set partitions of {1,2,3,4}, only {{1},{2},{3},{4}} has more than 3 blocks, so a(4) = 14.
G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 41*x^5 + 122*x^6 + 365*x^7 + ...

R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Per Alexandersson and Frether Getachew, An involution on derangements, arXiv:2105.08455 [math.CO], 2021.
N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 24 2012
V. Jelinek, T. Mansour, and M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, - From _N. J. A. Sloane_, Jan 01 2013
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243 [math.CO], 2012-2014 (Corollary 3, case k=4, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables, Transactions of the American Mathematical Society, 358 (2006), no. 1, 215-232.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A056272, A123183, A001519, A000110, A008277, A038754, A048328, A068911, A211216.

Essentially the same as A007051.

I:=[1, 1, 2]; [n le 3 select I[n] else  4*Self(n-1) - 3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 25 2012

a:= proc(n); if n<3 then [1,1,2][n+1]; else 4*a(n-1)-3*a(n-2); fi; end:
# Mike Zabrocki, Oct 25 2006
with(GraphTheory): G:=PathGraph(5): A:= AdjacencyMatrix(G): nmax:=27; for n from 0 to 2*nmax do B(n):=A^n; b(n):=B(n)[1,1]; od: for n from 0 to nmax do a(n):=b(2*n) od: seq(a(n),n=0..nmax);
# Johannes W. Meijer, May 29 2010

a=Exp[x]-1; Range[0, 20]! CoefficientList[Series[1+a+a^2/2+a^3/6, {x,0,20}],x]
Join[{1}, LinearRecurrence[{4, -3}, {1, 2}, 20]] (* David Nacin, Feb 26 2012 *)
CoefficientList[Series[1 / (1 - x / (1 - x / (1 - x / (1 - x)))), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 25 2012 *)
Table[Sum[StirlingS2[n,k],{k,0,3}],{n,0,30}] (* Robert A. Russell, Mar 29 2018 *)

{a(n) = if( n<1, n==0, (3^(n-1) + 1) / 2)}; /* Michael Somos, Apr 03 2014 */

def a(n, adict={0:1, 1:1, 2:2}):
    if n in adict:
        return adict[n]
    adict[n]=4*a(n-1) - 3*a(n-2)
    return adict[n] # David Nacin, Mar 04 2012

O.g.f.: (q^2 - 3*q + 1)/(3*q^2 - 4*q + 1) = Sum_{k=0..3} (q^k/Product_{i=1..k} (1-i*q)).
a(n) = 4*a(n-1) - 3*a(n-2); a(0) = 1, a(1) = 1, a(2) = 2, a(n) = Sum_{k=1..3} A008277(n,k).
Inverse binomial transform of A007581. - Philippe Deléham, Oct 30 2006
a(n) = Sum_{k=0..n} A056241(n,k), n >= 1. - Philippe Deléham, Oct 30 2006
a(0) = 1, a(n) = (3^(n-1) + 1)/2 for n >= 1, see A007051. - Philippe Deléham, Oct 30 2006
E.g.f.: (2 + 3*exp(x) + exp(3x))/6.
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x)))). - Michael Somos, May 03 2012
G.f.: 1 + x + 3*x^2*U(0)/2 where U(k) = 1 + 2/(3*3^k + 3*3^k/(1 - 18*x*3^k/ (9*x*3^k - 1/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012
G.f.: 1+x*G(0) where G(k) = 1 + 2*x/( 1-2*x - x*(1-2*x)/(x + (1-2*x)*2/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
a(n) = Sum_{k=0..3} Stirling2(n,k). - Robert A. Russell, Mar 29 2018
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=3. - Robert A. Russell, Apr 25 2018

A083878 a(0)=1, a(1)=3, a(n) = 6a(n-1) - 7a(n-2), n >= 2.

Original entry on oeis.org

1, 3, 11, 45, 193, 843, 3707, 16341, 72097, 318195, 1404491, 6199581, 27366049, 120799227, 533233019, 2353803525, 10390190017, 45864515427, 202455762443, 893682966669, 3944907462913, 17413664010795, 76867631824379

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A006012. Second binomial transform of A001333.

Third binomial transform of A077957. Inverse binomial transform of A083879. - Philippe Deléham, Dec 01 2008

Michael De Vlieger, Table of n, a(n) for n = 0..1551
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.
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A083879, A056241, A081179, A098158.

f[n_] := Simplify[(3 + Sqrt@2)^n + (3 - Sqrt@2)^n]/2; Array[f, 23, 0] (* Robert G. Wilson v, Oct 31 2010 *)

a(n) = ((3 - sqrt(2))^n + (3 + sqrt(2))^n)/2;
a(n) = Sum_{k=0..n} C(n, 2k)*3^(n-2k)*2^k;
G.f.: (1-3x)/(1-6x+7x^2);
E.g.f.: exp(3x)*cosh(x*sqrt(2)).
a(n) = Sum_{k=0..n} C(n, k)*2^((n-k)/2)(1+(-1)^(n-k))*3^k/2. - Paul Barry, Jan 22 2005
a(n) = Sum_{k=0..n} A098158(n,k)*3^(2k-n)*2^(n-k). - Philippe Deléham, Dec 01 2008
a(n) = A081179(n+1) - 3*A081179(n). - R. J. Mathar, Nov 10 2013
a(n) = Sum_{k=1..n} A056241(n, k) * 2^(k-1). - J. Conrad, Nov 23 2022

A180957 Generalized Narayana triangle for (-1)^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, -2, -5, -2, 1, 1, -5, -15, -15, -5, 1, 1, -9, -30, -41, -30, -9, 1, 1, -14, -49, -77, -77, -49, -14, 1, 1, -20, -70, -112, -125, -112, -70, -20, 1, 1, -27, -90, -126, -117, -117, -126, -90, -27, 1, 1, -35, -105, -90, 45, 131, 45, -90, -105, -35, 1

Offset: 0

Paul Barry, Sep 28 2010

			Triangle begins
  1;
  1,   1;
  1,   1,    1;
  1,   0,    0,    1;
  1,  -2,   -5,   -2,    1;
  1,  -5,  -15,  -15,   -5,    1;
  1,  -9,  -30,  -41,  -30,   -9,    1;
  1, -14,  -49,  -77,  -77,  -49,  -14,   1;
  1, -20,  -70, -112, -125, -112,  -70, -20,    1;
  1, -27,  -90, -126, -117, -117, -126, -90,  -27,   1;
  1, -35, -105,  -90,   45,  131,   45, -90, -105, -35, 1;

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

Cf. A056241, A139011.

Variant: A061176.

A180957:= func< n,k | (&+[ (-1)^(k-j)*Binomial(n, j)*Binomial(n-j, 2*(k-j)) : j in [0..n]]) >;
[A180957(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 06 2021

T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[n, j]*Binomial[n-j, 2*(k-j)], {j,0,n}];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)

def A180957(n,k): return sum( (-1)^(k+j)*binomial(n,j)*binomial(n-j, 2*(k-j)) for j in (0..n))
flatten([[A180957(n,k) for k in (0..n)] for n in [0..15]]) # G. C. Greubel, Apr 06 2021

G.f.: 1/(1 -x -x*y + x/(1 -x -x*y)) = (1 -x*(1+y))/(1 -2*x*(1+y) +x^2*(1 +3*y +y^2)).
E.g.f.: exp((1+y)*x) * cos(sqrt(y)*x).
T(n, k) = Sum_{j=0..n} (-1)^(k-j)*binomial(n,j)*binomial(n-j, 2*(k-j)).
Sum_{k=0..n} T(n, k) = A139011(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A180958(n) (diagonal sums).

A124216 Generalized Pascal triangle.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 34, 16, 1, 1, 25, 90, 90, 25, 1, 1, 36, 195, 328, 195, 36, 1, 1, 49, 371, 931, 931, 371, 49, 1, 1, 64, 644, 2240, 3334, 2240, 644, 64, 1, 1, 81, 1044, 4788, 9846, 9846

Offset: 0

Paul Barry, Oct 19 2006

Consider the 1-parameter family of triangles with g.f. (1-x(1+y))/(1-2x(1+y)+x^2(1+k*x+y^2)). A007318 corresponds to k=2. A056241 corresponds to k=1. A124216 corresponds to k=0. Row sums are A006012. Diagonal sums are A124217.

			Triangle begins
1,
1, 1,
1, 4, 1,
1, 9, 9, 1,
1, 16, 34, 16, 1,
1, 25, 90, 90, 25, 1,
1, 36, 195, 328, 195, 36, 1,
1, 49, 371, 931, 931, 371, 49, 1

P. Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5

Cf. A001263.

G.f.: (1-x(1+y))/(1-2x(1+y)+x^2(1+y^2)); Number triangle T(n,k)=sum{j=0..n, C(n,j)C(j,2(j-k))2^(j-k)}.

Equals 2*A001263 - A007318; (i.e. twice the Narayana triangle minus Pascal's triangle). - Gary W. Adamson, Jun 14 2007

A152265 a(n) = ((8 + sqrt(7))^n + (8 - sqrt(7))^n)/2.

Original entry on oeis.org

1, 8, 71, 680, 6833, 70568, 739607, 7811336, 82823777, 879934280, 9357993191, 99571637096, 1059740581649, 11280265991912, 120079042716599, 1278289521926600, 13608126915979457, 144867527905855112

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

nonn

Binomial transform of A145302. Inverse binomial transform of A152266. - Philippe Deléham, Dec 03 2008

Index entries for linear recurrences with constant coefficients, signature (16, -57).

Cf. A056241, A098158, A145302, A152266.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((8+r7)^n+(8-r7)^n)/2: n in [0..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 16*a(n-1) - 57*a(n-2), n > 1; a(0)=1, a(1)=8.
G.f.: (1-8*x)/(1-16*x+57*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*8^(2k-n)*7^(n-k). (End)
a(n) = Sum_{k=1..n} A056241(n,k) * 7^(k-1). - J. Conrad, Nov 23 2022

Extended beyond a(6) by Klaus Brockhaus, Dec 03 2008

A122935 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 19, 10, 1, 0, 1, 15, 45, 45, 15, 1, 0, 1, 21, 90, 141, 90, 21, 1, 0, 1, 28, 161, 357, 357, 161, 28, 1, 0, 1, 36, 266, 784, 1107, 784, 266, 36, 1, 0, 1, 45, 414, 1554, 2907, 2907, 1554, 414, 45, 1, 0, 1, 55, 615, 2850, 6765, 8953

Offset: 0

Philippe Deléham, Oct 30 2006

Subtriangle (1 <= k <= n) is in A056241.

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,   1;
  0, 1,  6,   6,    1;
  0, 1, 10,  19,   10,    1;
  0, 1, 15,  45,   45,   15,    1;
  0, 1, 21,  90,  141,   90,   21,    1;
  0, 1, 28, 161,  357,  357,  161,   28,    1;
  0, 1, 36, 266,  784, 1107,  784,  255,   36,   1;
  0, 1, 45, 414, 1554, 2907, 2907, 1554,  414,  45,  1;
  0, 1, 55, 615, 2850, 6765, 8953, 6765, 2850, 615, 55, 1;

Columns: A000007, A000012, A000217, A005712, A005714, A005716.

Cf. A027907, A056241.

T(2*k-1,k) = A082758(k-1)for k >= 1.
Sum_{k=0..n} T(n,k) = A124302(n); see also A007051.
Sum_{k=0..n} (-1)^(n-k)*T(n,k) = A117569(n).
G.f.: (1-x*(y+2)+x^2)/(1-2x*(1+y)+(1+y+y^2)*x^2). - Philippe Deléham, Oct 30 2011

cp's OEIS Frontend

A104027 Triangle, read by rows, equal to the matrix inverse of A056241, which is formed from the even-indexed trinomial coefficients.

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A104028 Column 1 of triangle A104027, which is the matrix inverse of the triangle A056241 of even-indexed trinomial coefficients.

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A124302 Number of set partitions with at most 3 blocks; number of Dyck paths of height at most 4; dimension of space of symmetric polynomials in 3 noncommuting variables.

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A083878 a(0)=1, a(1)=3, a(n) = 6*a(n-1) - 7*a(n-2), n >= 2.

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A180957 Generalized Narayana triangle for (-1)^n.

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A124216 Generalized Pascal triangle.

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A152265 a(n) = ((8 + sqrt(7))^n + (8 - sqrt(7))^n)/2.

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A122935 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A083878 a(0)=1, a(1)=3, a(n) = 6a(n-1) - 7a(n-2), n >= 2.