A211386 -id:A211386 - cp's OEIS frontend

A193844 Triangular array: the fission of ((x+1)^n) by ((x+1)^n); i.e., the self-fission of Pascal's triangle.

1, 1, 3, 1, 5, 7, 1, 7, 17, 15, 1, 9, 31, 49, 31, 1, 11, 49, 111, 129, 63, 1, 13, 71, 209, 351, 321, 127, 1, 15, 97, 351, 769, 1023, 769, 255, 1, 17, 127, 545, 1471, 2561, 2815, 1793, 511, 1, 19, 161, 799, 2561, 5503, 7937, 7423, 4097, 1023

Offset: 0

Clark Kimberling, Aug 07 2011

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
A193844 is also the fission of (p1(n,x)) by (q1(n,x)), where p1(n,x)=x^n+x^(n-1)+...+x+1 and q1(n,x)=(x+2)^n.
Essentially A119258 but without the main diagonal. - Peter Bala, Jul 16 2013
From Robert Coquereaux, Oct 02 2014: (Start)
This is also a rectangular array A(n,p) read down the antidiagonals:
1 1 1 1 1 1 1 1 1
3 5 7 9 11 13 15 17 19
7 17 31 49 71 97 127 161 199
15 49 111 209 351 545 799 1121 1519
31 129 351 769 1471 2561 4159 6401 9439
...
Calling Gr(n) the Grassmann algebra with n generators, A(n,p) is the dimension of the space of Gr(n)-valued symmetric multilinear forms with vanishing graded divergence. If p is odd A(n,p) is the dimension of the cyclic cohomology group of order p of the Z2 graded algebra Gr(n). If p is even, the dimension of this cohomology group is A(n,p)+1. A(n,p) = 2^n*A059260(p,n-1)-(-1)^p.
(End)
The n-th row are also the coefficients of the polynomial P=sum_{k=0..n} (X+2)^k (in falling order, i.e., that of X^n first). - M. F. Hasler, Oct 15 2014

			First six rows:
1
1....3
1....5....7
1....7....17....15
1....9....31....49....31
1....11...49....111...129...63

Jean-François Chamayou, A Random Difference Equation with Dufresne Variables revisited, arXiv:1410.1708 [math.PR], 2014.
R. Coquereaux and E. Ragoucy, Currents on Grassmann algebras, J. of Geometry and Physics, 1995, Vol 15, pp 333-352.
R. Coquereaux and E. Ragoucy, Currents on Grassmann algebras, arXiv:hep-th/9310147, 1993.
C. Kassel, A Künneth formula for the cyclic cohomology of Z2-graded algebras, Math. Ann. 275 (1986) 683.

Cf. A193842, A193845, A119258.
Columns, diagonals: A000225, A000337, A055580, A027608, A211386, A211388, A000012, A005408, A056220, A199899.
A145661 is an essentially identical triangle.

A193844 := (n,k) -> 2^k*binomial(n+1,k)*hypergeom([1,-k],[-k+n+2],1/2);
for n from 0 to 5 do seq(round(evalf(A193844(n,k))),k=0..n) od; # Peter Luschny, Jul 23 2014
# Alternatively
p := (n,x) -> add(x^k*(1+2*x)^(n-k), k=0..n): for n from 0 to 7 do [n], PolynomialTools:-CoefficientList(p(n,x), x) od; # Peter Luschny, Jun 18 2017

z = 10;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := (x + 1)^n
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193844 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193845 *)

# uses[fission from A193842]
p = lambda n,x: (x+1)^n
A193844_row = lambda n: fission(p, p, n)
for n in range(7): print(A193844_row(n)) # Peter Luschny, Jul 23 2014

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} (-1)^i*binomial(n+1,k-i)*2^(k-i).
O.g.f.: 1/( (1 - x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + 3*x)*t + (1 + 5*x + 7*x^2)*t^2 + ....
The n-th row polynomial R(n,x) = 1/(x+1)*( (2*x+1)^(n+1) - x^(n+1) ). (End)
T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - T(n-2,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 17 2014
T(n,k) = 2^k*binomial(n+1,k)*hyper2F1(1,-k,-k+n+2, 1/2). - Peter Luschny, Jul 23 2014

A193845 Mirror of the triangle A193844.

Original entry on oeis.org

1, 3, 1, 7, 5, 1, 15, 17, 7, 1, 31, 49, 31, 9, 1, 63, 129, 111, 49, 11, 1, 127, 321, 351, 209, 71, 13, 1, 255, 769, 1023, 769, 351, 97, 15, 1, 511, 1793, 2815, 2561, 1471, 545, 127, 17, 1, 1023, 4097, 7423, 7937, 5503, 2561, 799, 161, 19, 1

Offset: 0

Clark Kimberling, Aug 07 2011

This triangle is obtained by reversing the rows of the triangle A193844.
From Philippe Deléham, Jan 17 2014: (Start)
Subtriangle of the triangle in A112857.
T(n,0)   = A000225(n+1).
T(n,1)   = A000337(n).
T(n+2,2) = A055580(n).
T(n+3,3) = A027608(n).
T(n+4,4) = A211386(n).
T(n+5,5) = A211388(n).
T(n,n)   = A000012(n).
T(n+1,n) = A005408(n).
T(n+2,n) = A056220(n+2).
T(n+3,n) = A199899(n+1).
Row sums are A003462(n+1).
Diagonal sums are A048739(n).
Riordan array (1/((1-2*x)*(1-x)), x/(1-2*x)). (End)
Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-2)^0 + A_1*(x-2)^1 + A_2*(x-2)^2 + ... + A_n*(x-2)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0. - Derek Orr, Oct 14 2014
The n-th row lists the coefficients of the polynomial sum_{k=0..n} (X+2)^k, in order of increasing powers. - M. F. Hasler, Oct 15 2014

			First six rows:
1
3....1
7....5....1
15...17...7....1
31...49...31...9...1
63...129..111..49..11..1

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Russell Jay Hendel, A Method for Uniformly Proving a Family of Identities, arXiv:2107.03549 [math.CO], 2021.

Cf. A193844.
Cf. Columns: A000225, A000337, A055580, A027608, A211386, A211388
Cf. Diagonals: A000012, A005408, A056220, A199899

z = 10;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := (x + 1)^n
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193844 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193845 *)
Table[2^k*Binomial[n + 1, k]*Hypergeometric2F1[1, -k, -k + n + 2, 1/2], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Michael De Vlieger, Nov 09 2021 *)

for(n=0,20,for(k=0,n,print1(1/k!*sum(i=0,n,(2^(i-k)*prod(j=0,k-1,i-j))),", "))) \\ Derek Orr, Oct 14 2014

T(n,k) = A193844(n,n-k).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 17 2014

A211388 Expansion of 1/((1-2x)^6(1-x)).

Original entry on oeis.org

1, 13, 97, 545, 2561, 10625, 40193, 141569, 471041, 1496065, 4571137, 13516801, 38862849, 109051905, 299565057, 807600129, 2141192193, 5592842241, 14413725697, 36698062849, 92408905729, 230359564289, 568965726209, 1393398120449

Offset: 0

R. J. Mathar, Feb 07 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. H. Albert, M. D. Atkinson, R. Brignall, The enumeration of three pattern classes using monotone grid classes, El. J. Combinat. 19 (3) (2012) P20. Section 5.5.1
Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
Index entries for linear recurrences with constant coefficients, signature (13,-72,220,-400,432,-256,64).

Cf. A054849 (first differences).

m:=24; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)^6*(1-x)))); // Bruno Berselli, Feb 08 2013

CoefficientList[Series[1 / ((1 - 2 x)^6 (1 - x)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 10 2013 *)
LinearRecurrence[{13,-72,220,-400,432,-256,64},{1,13,97,545,2561,10625,40193},30] (* Harvey P. Dale, Sep 01 2023 *)

a(n) = 1 + 2^(n-2)*n*(n^4 + 10*n^3 + 55*n^2 + 110*n + 184)/15. - Bruno Berselli, Feb 08 2013

A211386(n) = a(n) - 2*a(n-1). - R. J. Mathar, Feb 08 2013

cp's OEIS Frontend

A193844 Triangular array: the fission of ((x+1)^n) by ((x+1)^n); i.e., the self-fission of Pascal's triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A193845 Mirror of the triangle A193844.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A211388 Expansion of 1/((1-2x)^6(1-x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

cp's OEIS Frontend

A193844 Triangular array: the fission of ((x+1)^n) by ((x+1)^n); i.e., the self-fission of Pascal's triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A193845 Mirror of the triangle A193844.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A211388 Expansion of 1/((1-2*x)^6*(1-x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A211388 Expansion of 1/((1-2x)^6(1-x)).