A104027 -id:A104027 - cp's OEIS frontend

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

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])}

A006846 Hammersley's polynomial p_n(1).

Original entry on oeis.org

1, 1, 2, 7, 41, 376, 5033, 92821, 2257166, 69981919, 2694447797, 126128146156, 7054258103921, 464584757637001, 35586641825705882, 3136942184333040727, 315295985573234822561, 35843594275585750890976, 4575961401477587844760793, 651880406652100451820206941

Offset: 0

N. J. A. Sloane

nonn

Equals column 0 of triangle A104027. Also equals column 0 of triangle A104030 (offset 1). Both A104027 and A104030 involve the trinomial coefficients. - Paul D. Hanna, Mar 06 2005

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..100
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)

Cf. A065547, A085707, A104027, A104030.

function A006846list(len::Int)  # Algorithm of L. Seidel (1877)
    R = Array{BigInt}(len)
    A = fill(BigInt(0), len+1); A[1] = 1
    for n in 1:len
        for k in n:-1:2 A[k] += A[k+1] end
        for k in 2:1:n A[k] += A[k-1] end
        R[n] = A[n]
    end
    return R
end
println(A006846list(20)) # Peter Luschny, Jan 02 2018

A006846 := proc(n)
    option remember ;
    if n =0 then
        return 1;
    else
        add(binomial(2*n,2*m)*procname(m)/(-4)^(n-m),m=0..n-1) ;
        (3/4)^n-% ;
    end if
end proc:
seq(A006846(n),n=0..20) ; # R. J. Mathar, Jan 10 2018

h[n_, x_] := Sum[c[k] x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x-1)] == EulerE[2*n, x], x]; a[n_] := Sum[(-1)^(n+k)*c[k], {k, 0, n}] /. eq[n] // First; Table[a[n], {n, 0, 15}]   (* Jean-François Alcover, Oct 02 2013, after Philippe Deléham *)

{a(n)=local(X=x+x*O(x^(2*n))); round((2*n)!*polcoeff(cosh(sqrt(3)*X/2)/cos(X/2),2*n))} \\ Paul D. Hanna

a(n) = Sum_{k>=0} (-1)^(n+k)*A065547(n, k) = Sum_{k>=0} A085707(n, k). - Philippe Deléham, Feb 26 2004
E.g.f.: cosh(sqrt(3)*x/2)/cos(x/2) = Sum_{n>=0} a(n)*x^(2n)/(2n)!. - Paul D. Hanna, Feb 27 2005
a(n) = (-1)^n*A104027(n, 0). a(n+1) = (-1)^(n+1)*A104030(n, 0). - Paul D. Hanna, Mar 06 2005
G.f.: 1/(1-x/(1-x/(1-3x/(1-4x/(1-7x/(1-.../(1-ceiling((n+1)^2/4)*x/(1-... (continued fraction). - Paul Barry, Feb 24 2010
a(n) ~ 4*cosh(sqrt(3)*Pi/2) * (2*n)! / Pi^(2*n+1). - Vaclav Kotesovec, Jun 07 2021

A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

Original entry on oeis.org

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

Offset: 1

Colin Mallows, Aug 23 2000

Forms the even-indexed trinomial coefficients (A027907). Matrix inverse is A104027. - Paul D. Hanna, Feb 26 2005

Subtriangle (for 1<=k<=n) of triangle defined by [0, 1, 0, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 29 2006

			Triangle begins:
  1;
  1,1;
  1,3,1;
  1,6,6,1;
  1,10,19,10,1;
  ...
Triangle (0, 1, 0, 1, 0, 0, 0...) DELTA (1, 0, 1, 0, 0, 0, ...) 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;
  ... - _Philippe Deléham_, Mar 27 2014

Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 6.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 7.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.

Columns are A000217, A005712, A005714, A005716.

Cf. A027907, A104027, A117569, A124302.

t[n_, k_] := Sum[ Binomial[n, j]*Binomial[n-j, 2*(k-j)], {j, 0, n}]; Flatten[ Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Oct 11 2011, after Paul Barry *)

PARI
```
T(n,k)=if(nPaul D. Hanna
```

T(n, k) = Sum_{j=0..k-1} C(n-1, 2k-j-2)*C(2k-j-2, j).
G.f.: A(x, y) = (1 - x*(1+y))/(1 - 2*x*(1+y) + x^2*(1+y+y^2)) (offset=0). - Paul D. Hanna, Feb 26 2005
Sum_{k, 1<=k<=n}T(n,k)=A124302(n). Sum_{k, 1<=k<=n}(-1)^(n-k)*T(n,k)=A117569(n). - Philippe Deléham, Oct 29 2006
From Paul Barry, Sep 28 2010: (Start)
G.f.: 1/(1-x-xy-x^2y/(1-x-xy)).
E.g.f.: exp((1+y)x)*cosh(sqrt(y)*x).
T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2*(k-j)). (End)
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,1) = T(2,1) = T(2,2) = 1, T(n,k) = 0 if k<1 or if k>n. - Philippe Deléham, Mar 27 2014

More terms from James Sellers, Aug 25 2000

More terms from Paul D. Hanna, Feb 26 2005

A104030 Matrix inverse, read by rows, of triangle A104029, which forms the pairwise sums of trinomial coefficients.

Original entry on oeis.org

1, -2, 1, 7, -5, 1, -41, 32, -9, 1, 376, -299, 91, -14, 1, -5033, 4015, -1241, 205, -20, 1, 92821, -74080, 22954, -3842, 400, -27, 1, -2257166, 1801537, -558402, 93652, -9863, 707, -35, 1, 69981919, -55855829, 17313721, -2904530, 306409, -22190, 1162, -44, 1, -2694447797, 2150565968

Offset: 0

Paul D. Hanna, Feb 26 2005

Column 0 forms signed Hammersley's polynomial p_n(1) (A006846), offset 1.
Row sums equal negative Genocchi numbers of first kind (A001469).
Rows form polynomials R_n(x) such that: R_n(3) = 1 for n>=0 and R_n(1/2) = (-1)^n*A005647(n+1)/2^n (signed Salie numbers).
Column 1 forms A104031.
Unsigned row sums form A104032.

			Rows begin:
1;
-2,1;
7,-5,1;
-41,32,-9,1;
376,-299,91,-14,1;
-5033,4015,-1241,205,-20,1;
92821,-74080,22954,-3842,400,-27,1;
-2257166,1801537,-558402,93652,-9863,707,-35,1; ...

Cf. A006846, A001469, A005647, A104027, A104029, A104031, A104032.

T(n,k)=if(n=j, polcoeff((1+x+x^2)^(m-1)+O(x^(2*j)),2*j-2)+ polcoeff((1+x+x^2)^(m-1)+O(x^(2*j)),2*j-1))))^-1)[n+1,k+1])

A104029 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 13, 9, 1, 5, 26, 35, 14, 1, 6, 45, 96, 75, 20, 1, 7, 71, 216, 267, 140, 27, 1, 8, 105, 427, 750, 623, 238, 35, 1, 9, 148, 770, 1800, 2123, 1288, 378, 44, 1, 10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1, 11, 265, 2067, 7590, 15115, 17303, 11505

Offset: 0

Paul D. Hanna, Feb 26 2005

Matrix inverse is A104030. Antidiagonal sums form unsigned A078039.

			Row 3: {4,13,9,1} is formed from the pairwise sums
of row 3 of A027907: {1,3, 6,7, 6,3, 1}.
Rows begin:
1;
2, 1;
3, 5, 1;
4, 13, 9, 1;
5, 26, 35, 14, 1;
6, 45, 96, 75, 20, 1;
7, 71, 216, 267, 140, 27, 1;
8, 105, 427, 750, 623, 238, 35, 1;
9, 148, 770, 1800, 2123, 1288, 378, 44, 1;
10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1;
11, 265, 2067, 7590, 15115, 17303, 11505, 4302, 825, 65, 1;
12, 341, 3157, 13959, 34210, 49721, 43923, 23397, 7194, 1155, 77, 1; ...

Cf. A104030, A104027, A078039.

{T(n,k)=polcoeff((1+x+x^2)^n+x*O(x^(2*k)),2*k)+ polcoeff((1+x+x^2)^n+x*O(x^(2*k+1)),2*k+1)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

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

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

T(n, k) = [x^(2k)](1+x+x^2)^n + [x^(2k+1)](1+x+x^2)^n.

cp's OEIS Frontend

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

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A006846 Hammersley's polynomial p_n(1).

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A056241 Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).

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A104030 Matrix inverse, read by rows, of triangle A104029, which forms the pairwise sums of trinomial coefficients.

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A104029 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).

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