A193649 -id:A193649 - cp's OEIS frontend

A193722 Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.

1, 1, 2, 1, 5, 6, 1, 8, 21, 18, 1, 11, 45, 81, 54, 1, 14, 78, 216, 297, 162, 1, 17, 120, 450, 945, 1053, 486, 1, 20, 171, 810, 2295, 3888, 3645, 1458, 1, 23, 231, 1323, 4725, 10773, 15309, 12393, 4374, 1, 26, 300, 2016, 8694, 24948, 47628, 58320, 41553, 13122

Offset: 0

Clark Kimberling, Aug 04 2011

Suppose that p = p(n)*x^n + p(n-1)*x^(n-1) + ... + p(1)*x + p(0) is a polynomial and that Q is a sequence of polynomials
...
q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k),
...
for k=0,1,2,...  The Q-upstep of p is the polynomial given by
...
U(p) = p(n)*q(n+1,x) + p(n-1)*q(n,x) + ... + p(0)*q(1,x); note that q(0,x) does not appear.
...
Now suppose that P=(p(n,x)) and Q=(q(n,x)) are sequences of polynomials, where n indicates degree.  The fusion of P by Q, denoted by P**Q, is introduced here as the sequence W=(w(n,x)) of polynomials defined by w(0,x)=1 and w(n+1,x)=U(p(n,x)).
...
Strictly speaking, ** is an operation on sequences of polynomials.  However, if P and Q are regarded as numerical triangles (e.g., coefficients of polynomials), then ** can be regarded as an operation on numerical triangles.  In this case, row (n+1) of P**Q, for n >= 0, is given by the matrix product P(n)*QQ(n), where P(n)=(p(n,n)...p(n,n-1)......p(n,1), p(n,0)) and QQ(n) is the (n+1)-by-(n+2) matrix given by
...
q(n+1,0) .. q(n+1,1)........... q(n+1,n) .... q(n+1,n+1)
0 ......... q(n,0)............. q(n,n-1) .... q(n,n)
0 ......... 0.................. q(n-1,n-2) .. q(n-1,n-1)
...
0 ......... 0.................. q(2,1) ...... q(2,2)
0 ......... 0 ................. q(1,0) ...... q(1,1);
here, the polynomial q(k,x) is taken to be
q(k,0)*x^k + q(k,1)x^(k-1) + ... + q(k,k)*x+q(k,k-1); i.e., "q" is used instead of "t".
...
If s=(s(1),s(2),s(3),...) is a sequence, then the infinite square matrix indicated by
s(1)...s(2)...s(3)...s(4)...s(5)...
..0....s(1)...s(2)...s(3)...s(4)...
..0......0....s(1)...s(2)...s(3)...
..0......0.......0...s(1)...s(2)...
is the self-fusion matrix of s; e.g., A202453, A202670.
...
Example:  let p(n,x)=(x+1)^n and q(n,x)=(x+2)^n.  Then
  ...
w(0,x) = 1 by definition of W
w(1,x) = U(p(0,x)) = U(1) = p(0,0)*q(1,x) = 1*(x+2) = x+2;
w(2,x) = U(p(1,x)) = U(x+1) = q(2,x) + q(1,x) = x^2+5x+6;
w(3,x) = U(p(2,x)) = U(x^2+2x+1) = q(3,x) + 2q(2,x) + q(1,x) = x^3+8x^2+21x+18;
  ...
From these first 4 polynomials in the sequence P**Q, we can write the first 4 rows of P**Q when P, Q, and P**Q are regarded as triangles:
  1;
  1,  2;
  1,  5,  6;
  1,  8, 21, 18;
  ...
Generally, if P and Q are the sequences given by p(n,x)=(ax+b)^n and q(n,x)=(cx+d)^n, then P**Q is given by (cx+d)(bcx+a+bd)^n.
...
In the following examples, r(P**Q) is the mirror of P**Q, obtained by reversing the rows of P**Q.
...
..P...........Q.........P**Q.......r(P**Q)
(x+1)^n.....(x+1)^n.....A081277....A118800 (unsigned)
(x+1)^n.....(x+2)^n.....A193722....A193723
(x+2)^n.....(x+1)^n.....A193724....A193725
(x+2)^n.....(x+2)^n.....A193726....A193727
(x+2)^n.....(2x+1)^n....A193728....A193729
(2x+1)^n....(x+1)^n.....A038763....A136158
(2x+1)^n....(2x+1)^n....A193730....A193731
(2x+1)^n,...(x+1)^n.....A193734....A193735
...
Continuing, let u denote the polynomial x^n+x^(n-1)+...+x+1, and let Fibo[n,x] denote the n-th Fibonacci polynomial.
...
P.............Q.........P**Q.......r(P**Q)
Fib[n+1,x]...(x+1)^n....A193736....A193737
u.............u.........A193738....A193739
u**u..........u**u......A193740....A193741
...
Regarding A193722:
col 1 ..... A000012
col 2 ..... A016789
col 3 ..... A081266
w(n,n) .... A025192
w(n,n-1) .. A081038
...
Associated with "upstep" as defined above is "downstep" defined at A193842 in connection with fission.

			First six rows:
  1;
  1,   2;
  1,   5,   6;
  1,   8,  21,  18;
  1,  11,  45,  81,  54;
  1,  14,  78, 216, 297, 162;

Jinyuan Wang, Rows n=0..101 of triangle, flattened
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Cf. A081277, A084938, A118800, A193649, A193723-A193741, A202453.

Flat(List([0..10], n-> List([0..n], k-> 3^(k-1)*( Binomial(n-1,k) + 2*Binomial(n,k) ) ))); # G. C. Greubel, Feb 18 2020

[3^(k-1)*( Binomial(n-1,k) + 2*Binomial(n,k) ): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

fusion := proc(p, q, n) local d, k;
p(n-1,0)*q(n,x)+add(coeff(p(n-1,x),x^k)*q(n-k,x), k=1..n-1);
[1,seq(coeff(%,x,n-1-k), k=0..n-1)] end:
p := (n, x) -> (x + 1)^n; q := (n, x) -> (x + 2)^n;
A193722_row := n -> fusion(p, q, n);
for n from 0 to 5 do A193722_row(n) od; # Peter Luschny, Jul 24 2014

(* First program *)
z = 9; a = 1; b = 1; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, 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}]] (* A193722 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)
(* Second program *)
Table[3^(k-1)*(Binomial[n-1,k] +2*Binomial[n,k]), {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)

T(n,k) = 3^(k-1)*(binomial(n-1,k) +2*binomial(n,k)); \\ G. C. Greubel, Feb 18 2020

def fusion(p, q, n):
    F = p(n-1,0)*q(n,x)+add(expand(p(n-1,x)).coefficient(x,k)*q(n-k,x) for k in (1..n-1))
    return [1]+[expand(F).coefficient(x,n-1-k) for k in (0..n-1)]
A193842_row = lambda k: fusion(lambda n,x: (x+1)^n, lambda n,x: (x+2)^n, k)
for n in range(7): A193842_row(n) # Peter Luschny, Jul 24 2014

Triangle T(n,k), read by rows, given by [1,0,0,0,0,0,0,0,...] DELTA [2,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011
T(n,k) = 3*T(n-1,k-1) + T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
T(n, k) = 3^(k-1)*( binomial(n-1,k) + 2*binomial(n,k) ). - G. C. Greubel, Feb 18 2020

A193842 Triangular array: the fission of the polynomial sequence ((x+1)^n: n >= 0) by the polynomial sequence ((x+2)^n: n >= 0). (Fission is defined at Comments.)

Original entry on oeis.org

1, 1, 4, 1, 7, 13, 1, 10, 34, 40, 1, 13, 64, 142, 121, 1, 16, 103, 334, 547, 364, 1, 19, 151, 643, 1549, 2005, 1093, 1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1, 28, 349, 2542, 11926, 37384, 78322, 105796

Offset: 0

Clark Kimberling, Aug 07 2011

Suppose that p = p(n)*x^n + p(n-1)*x^(n-1) + ... + p(1)*x + p(0) is a polynomial and that Q is a sequence of polynomials:
...
q(k,x) = t(k,0)*x^k + t(k,1)*x^(k-1) + ... + t(k,k-1)*x + t(k,k),
...
for k = 0, 1, 2, ...  The Q-downstep of p is the polynomial given by
...
D(p) = p(n)*q(n-1,x) + p(n-1)*q(n-2,x) + ... + p(1)*q(0,x). (Note that p(0) does not appear. "Q-downstep" as just defined differs slightly from "Q-downstep" as defined for a different purpose at A193649.)
...
Now suppose that P = (p(n,x): n >= 0) and Q = (q(n,x): n >= 0) are sequences of polynomials, where n indicates degree.  The fission of P by Q, denoted by P^^Q, is introduced here as the sequence W = (w(n,x): n >= 0) of polynomials defined by w(0,x) = 1 and w(n,x) = D(p(n+1,x)).
...
Strictly speaking, ^^ is an operation on sequences of polynomials.  However, if P and Q are regarded as numerical triangles (of coefficients of polynomials), then ^^ can be regarded as an operation on numerical triangles.  In this case, row n of P^^Q, for n > 0, is given by the matrix product P(n+1)*QQ(n), where P(n+1) =(p(n+1,n+1), p(n+1,n), ..., p(n+1,2), p(n+1,1)) and QQ(n) is the (n+1)-by-(n+1) matrix given by
...
q(n,0) .. q(n,1)............. q(n,n-1) .... q(n,n)
0 ....... q(n-1,0)........... q(n-1,n-2)... q(n-1,n-1)
0 ....... 0.................. q(n-2,n-3) .. q(n-2,n-2)
...
0 ....... 0.................. q(1,0) ...... q(1,1)
0 ....... 0 ................. 0 ........... q(0,0).
Here, the polynomial q(k,x) is taken to be
q(k,0)*x^k + q(k,1)x^(k-1) + ... + q(k,k)*x + q(k,k);
i.e., "q" is used instead of "t".
...
Example:  Let p(n,x) = (x+1)^n and q(n,x) = (x+2)^n.  Then
...
w(0,x) = 1 by the definition of W,
w(1,x) = D(p(2,x)) = 1*(x+2) + 2*1 = x + 4,
w(2,x) = D(p(3,x)) = 1*(x^2+4*x+4) + 3*(x+2) + 3*1 = x^2 + 7*x + 13,
w(3,x) = D(p(4,x)) = 1*(x^3+6*x^2+12*x+8) + 4*(x^2+4x+4) + 6*(x+2) + 4*1 = x^3 + 10*x^2 + 34*x + 40.
...
From these first 4 polynomials in the sequence P^^Q, we can write the first 4 rows of P^^Q when P, Q, and P^^Q are regarded as triangles:
1
1...4
1...7....13
1...10...34...40
...
In the following examples, r(P^^Q) is the mirror of P^^Q, obtained by reversing the rows of P^^Q.  Let u denote the polynomial x^n + x^(n-1) + ... + x + 1.
...
..P........Q...........P^^Q........r(P^^Q)
(x+1)^n....(x+2)^n.....A193842.....A193843
(x+1)^n....(x+1)^n.....A193844.....A193845
(x+2)^n....(x+1)^n.....A193846.....A193847
(2x+1)^n...(x+1)^n.....A193856.....A193857
(x+1)^n....(2x+1)^n....A193858.....A193859
(x+1)^n.......u........A054143.....A104709
..u........(x+1)^n.....A074909.....A074909
..u...........u........A002260.....A004736
(x+2)^n.......u........A193850.....A193851
..u.........(x+2)^n....A193844.....A193845
(2x+1)^n......u........A193860.....A193861
..u.........(2x+1)^n...A115068.....A193862
...
Regarding A193842,
col 1 ...... A000012
col 2 ...... A016777
col 3 ...... A081271
w(n,n) ..... A003462
w(n,n-1) ... A014915

			First six rows, for 0 <= k <= n and 0 <= n <= 5:
  1
  1...4
  1...7....13
  1...10...34....40
  1...13...64....142...121
  1...16...103...334...547...364

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Digital Library of Mathematical Functions, Hypergeometric function, analytic properties.
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52(3) (2014), 195-202.

Cf. A193722 (fusion of P by Q), A193649 (Q-residue), A193843 (mirror of A193842).

[ (&+[3^(k-j)*Binomial(n-j,k-j): j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

fission := proc(p, q, n) local d, k;
p(n+1,0)*q(n,x)+add(coeff(p(n+1,x),x^k)*q(n-k,x), k=1..n);
seq(coeff(%,x,n-k), k=0..n) end:
A193842_row := n -> fission((n,x) -> (x+1)^n, (n,x) -> (x+2)^n, n);
for n from 0 to 5 do A193842_row(n) od; # Peter Luschny, Jul 23 2014
# Alternatively:
p := (n,x) -> add(x^k*(1+3*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

(* First program *)
z = 10;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := (x + 2)^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}]]  (* A193842 *)
TableForm[Table[h[n], {n, 0, z}]]  (* A193843 *)
Flatten[Table[h[n], {n, -1, z}]]
(* Second program *)
Table[SeriesCoefficient[((x+3)^(n+1) -1)/(x+2), {x,0,n-k}], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)

T(n,k) = sum(j=0,k, 3^(k-j)*binomial(n-j,k-j)); \\ G. C. Greubel, Feb 18 2020

from mpmath import mp, hyp2f1
mp.dps = 100; mp.pretty = True
def T(n,k):
    return 3^k*binomial(n,k)*hyp2f1(1,-k,-n,1/3)-0^(n-k)//2
for n in range(7):
    print([int(T(n,k)) for k in (0..n)]) # Peter Luschny, Jul 23 2014

# Second program using the 'fission' operation.
def fission(p, q, n):
    F = p(n+1,0)*q(n,x)+add(expand(p(n+1,x)).coefficient(x,k)*q(n-k,x) for k in (1..n))
    return [expand(F).coefficient(x,n-k) for k in (0..n)]
A193842_row = lambda k: fission(lambda n,x: (x+1)^n, lambda n,x: (x+2)^n, k)
for n in range(7): A193842_row(n) # Peter Luschny, Jul 23 2014

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = Sum_{i = 0..k} 3^(k-i)*binomial(n-i,k-i).
O.g.f.: 1/((1 - x*t)*(1 - (1 + 3*x)*t)) = 1 + (1 + 4*x)*t + (1 + 7*x + 13*x^2)*t^2 + ....
The n-th row polynomial is R(n,x) = (1/(2*x + 1))*((3*x + 1)^(n+1) - x^(n+1)). (End)
T(n,k) = T(n-1,k) + 4*T(n-1,k-1) - T(n-2,k-1) - 3*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 4, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 17 2014
T(n,k) = 3^k * C(n,k) * hyp2F1(1, -k, -n, 1/3) with or without the additional term -0^(n-k)/2 depending on the exact definition of the hypergeometric function used. Compare formulas 15.2.5 and 15.2.6 in the DLMF reference. - Peter Luschny, Jul 23 2014

Name and Comments edited by Petros Hadjicostas, Jun 05 2020

A090040 (3*6^n + 2^n)/4.

Original entry on oeis.org

1, 5, 28, 164, 976, 5840, 35008, 209984, 1259776, 7558400, 45349888, 272098304, 1632587776, 9795522560, 58773127168, 352638746624, 2115832446976, 12694994616320, 76169967566848, 457019805138944, 2742118830309376

Offset: 0

Paul Barry, Nov 20 2003

A090040 is the Q-residue of the triangle A175840, where Q is the triangular array (t(i,j)) given by t(i,j)=1; see A193649 for the definition of Q-residue. - Clark Kimberling, Aug 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Cf. A081335.

[(3*6^n+2^n)/4: n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

LinearRecurrence[{8,-12},{1,5},30] (* Harvey P. Dale, Nov 23 2014 *)

a(n)=(3*6^n+2^n)/4 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1-3*x)/((1-2*x)*(1-6*x)).
E.g.f.: (3*exp(6*x)+exp(2*x))/4 = exp(4*x)*(cosh(2*x)+sinh(2*x)/2).
a(n) = 8*a(n-1) -12*a(n-2), a(0)=1, a(1)=5.
a(n) = (3*6^n+2^n)/4.
a(n)=6*a(n-1)-2^(n-1). - Paul Curtz, Jan 09 2009
Fourth binomial transform of (1, 1, 4, 4, 16, 16, ...). a(n)=sum{k=1..floor(n/2), C(n, 2k)4^(n-k-1)}. - Paul Barry, Nov 22 2003
a(n) = A019590 (mod 4), proof via a(n)=8*a(n-1)-12*a(n-2). - R. J. Mathar, Feb 25 2009
a(n) = Sum_{k, 0<=k<=n} A117317(n,k)*3^k. - Philippe Deléham, Jan 28 2012

A193651 a(n) = ((2*n + 1)!! + 1)/2.

Original entry on oeis.org

1, 2, 8, 53, 473, 5198, 67568, 1013513, 17229713, 327364538, 6874655288, 158117071613, 3952926790313, 106729023338438, 3095141676814688, 95949391981255313, 3166329935381425313, 110821547738349885938, 4100397266318945779688, 159915493386438885407813

Offset: 0

Clark Kimberling, Aug 02 2011

Previous name was: Q-residue of the triangle A130534, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.

a(404) has 1002 decimal digits. - Michael De Vlieger, Apr 25 2016

Michael De Vlieger, Table of n, a(n) for n = 0..403

Cf. A001147, A193649, A130534.

seq((1+doublefactorial(2*n+1))/2,n=0..18); # Peter Luschny, Aug 20 2014

q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
u[0, x_] := 1; u[n_, x_] := (x + n)*u[n - 1, x]
p[n_, k_] := Coefficient[u[n, x], x, k]
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}]    (* A193651 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]  (* A130534 *)
Table[((2 n + 1)!! + 1)/2, {n, 0, 18}] (* or *)
Table[(2^n Gamma[n + 3/2])/Sqrt[Pi] + 1/2, {n, 0, 18}] (* or *)
Table[2^n Pochhammer[1/2, n + 1] + 1/2, {n, 0, 18}] (* Michael De Vlieger, Apr 25 2016 *)

def A():
    n, a, b = 1, 1, 2
    yield a
    while True:
        yield b
        n += 1
        a, b = b, ((2*(b-a)*n + a)*n - b)/(n-1)
A193651 = A()
[next(A193651) for i in range(19)] # Peter Luschny, Aug 20 2014

From Peter Luschny, Aug 20 2014: (Start)
a(n) = (2^n*Gamma(n+3/2))/sqrt(Pi) + 1/2.
a(n) = 2^n*Pochhammer(1/2, n+1) + 1/2.
a(n) = ((2*a(n-1) - 2*a(n-2))*n^2 + a(n-2)*n - a(n-1))/(n-1) for n>1, a(0)=1, a(1)=2. (End)
(-n+1)*a(n) +(2*n^2-1)*a(n-1) -n*(2*n-1)*a(n-2)=0. - R. J. Mathar, Feb 19 2015
E.g.f.: (exp(x) + 1/(1-2*x)^(3/2))/2. - Vladimir Reshetnikov, Apr 25 2016

New name from Peter Luschny, Aug 20 2014

A193657 First difference of A002627.

Original entry on oeis.org

1, 2, 7, 31, 165, 1031, 7423, 60621, 554249, 5611771, 62353011, 754471433, 9876716941, 139097096919, 2097156230471, 33704296561141, 575219994643473, 10389911153247731, 198019483156015579, 3971390745517868001, 83608226221428800021, 1843561388182505040463

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

Previous name was: Q-residue of the triangle A094727, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.
Number of n X n rook placements avoiding the pattern 001. - N. J. A. Sloane, Feb 04 2013
Let M(n) denote the n X n matrix with ones along the subdiagonal, ones everywhere above the main diagonal, the integers 2, 3, etc., along the main diagonal, and zeros everywhere else. Then a(n) is equal to the permanent of M(n). - John M. Campbell, Apr 20 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013

Cf. A193649, A094727, A082425, A114870, A176305, A296964.

a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1,1):
seq(simplify(a(n)), n=0..9); # Peter Luschny, May 30 2014

q[n_, k_] := n + k + 1;  (* A094727 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := 1
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}]    (* A193657 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A193668 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]
CoefficientList[Series[(E^x-x)/(x-1)^2,{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Nov 20 2012 *)

a(n) = { sum(k=0, n, if (k <= n-2, binomial(n,k)*(k+1)!, binomial(n,k)^2*k!));} \\ Michel Marcus, Feb 07 2013

def A193657():
    a = 2; b = 7; c = 31; n = 3
    yield 1
    while True:
        yield a
        n += 1
        a,b,c = b,c,((n-2)^2*a+2*(1+n-n^2)*b+(3*n+n^2-2)*c)/n
a = A193657(); [next(a) for n in range(19)] # Peter Luschny, May 30 2014

E.g.f.: (exp(x)-x)/(x-1)^2. - Vaclav Kotesovec, Nov 20 2012
a(n) ~ n!*n*(e-1). - Vaclav Kotesovec, Nov 20 2012
a(n) = 1-n*Gamma(n+1)+e*n*Gamma(n+1,1). - Peter Luschny, May 30 2014
a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, May 30 2014
From Peter Bala, Feb 10 2020: (Start)
a(n) = n*A002627(n) + 1.
a(n) = A114870(n) + n!.
a(n) = A296964(n+1) - A296964(n) for n >= 2.
a(1) = 2 and a(n) = (n^2*a(n-1) - 1)/(n - 1) for n >= 2. See A082425 for solutions to this recurrence with different starting values.
Also, a(0) = 1 and a(n) = n*( a(n-1) + ... + a(0) ) + 1 for n >= 1.
Second column of A176305. (End)

Simpler definition by Peter Luschny, May 30 2014

A193664 Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q=Pascal's triangle. (See Comments.)

Original entry on oeis.org

0, 1, 1, 6, 11, 68, 177, 1215, 4059, 30733, 124408, 1027972, 4862600, 43450761, 234283662, 2247091674, 13563976285, 138780931929, 925063455844, 10044476018973, 73144254450840, 839146997933059, 6618306039456419

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

The definition of Q-residue is given at A193649.

Cf. A049310, A193649, A000110.

f[n_, x_] := Fibonacci[n, x];
q[n_, k_] := Coefficient[(x + 1)^n, x, k]; (* Pascal's triangle *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
p[n_, k_] := Coefficient[f[n, x], x, k];
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 22}]    (* A193664 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A000110 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

A193650 Q-residue of the coefficient triangle (A076699) of the cyclotomic polynomials, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

Original entry on oeis.org

1, 0, 3, 7, 10, 31, 24, 127, 136, 292, 352, 2047, 1664, 8191, 5504, 9664, 32896, 131071, 116736, 524287, 419840, 603904, 1398784, 8388607, 7897088, 17318416

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

For the definition of Q-residue, see A193649.

Cf. A193649, A076699.

q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := Coefficient[Cyclotomic[n, x], x, k]
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 25}]    (* A193650 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 8}, {k, 0, n}]]  (* A076699 *)

A193653 Q-residue of the Delannoy triangle A008288, where Q is the triangular array (t(i,j)) given by t(i,j)=1.

Original entry on oeis.org

1, 2, 6, 20, 70, 248, 882, 3140, 11182, 39824, 141834, 505148, 1799110, 6407624, 22821090, 81278516, 289477726, 1030990208, 3671926074, 13077758636, 46577128054, 165886901432, 590814960402, 2104218684068, 7494285973006, 26691295287152, 95062457807466

Offset: 0

Clark Kimberling, Aug 02 2011

For the definition of Q-residue, see A193649.

This sequence gives the number of closed walks from the two vertices having loops in the digraph defined by its adjacency matrix A = (2,1,1; 1,2,1; 1,1,0). - David Neil McGrath, Aug 22 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-2).

Cf. A008288, A193649.

[n le 3 select Factorial(n) else 4*Self(n-1) -Self(n-2) -2*Self(n-3): n in [1..41]]; // G. C. Greubel, May 25 2021

(* First program *)
q[n_, k_] := 1;
r[0] = 1; r[k_]:= Sum[q[k-1, i]*r[k-1-i], {i, 0, k-1}]
p[n_, k_]:= p[n, k]= If[k==0 || k==n, 1, p[n-1, k-1] + p[n-2, k-1] + p[n-1, k]];  (* A008288, Delannoy *)
v[n_]:= Sum[p[n, k]*r[n-k], {k, 0, n}];
Table[v[n], {n, 0, 16}]    (* A193653 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
(* Second program *)
LinearRecurrence[{4,-1,-2}, {1,2,6}, 40] (* G. C. Greubel, May 25 2021 *)

Vec((1-2*t-t^2)/(1-4*t+t^2+2*t^3) + O(t^40)) \\ Michel Marcus, Aug 23 2014

a(n) = round((34+(17-3*sqrt(17))*((3-sqrt(17))/2)^n+((3+sqrt(17))/2)^n*(17+3*sqrt(17)))/68) \\ Colin Barker, Sep 02 2016

[(1/2)*(1 + sum(binomial(n-k,k)*2^k*3^(n-2*k) for k in (0..n//2))) for n in (0..40)] # G. C. Greubel, May 25 2021

From David Neil McGrath, Aug 22 2014: (Start)
a(n) = 4*a(n-1) - a(n-2) - 2*a(n-3).
a(n-1) = (1,1) and (2,2) elements of A^(n-1) where A=(2,1,1; 1,2,1; 1,1,0) and n>1. (End)
G.f.: (1-2*t-t^2)/(1-4*t+t^2+2*t^3). - Robert Israel, Aug 22 2014
a(n) = (34 + (17-3*sqrt(17))*((3-sqrt(17))/2)^n + ((3+sqrt(17))/2)^n*(17+3*sqrt(17)))/68. - Colin Barker, Sep 02 2016
From G. C. Greubel, May 25 2021: (Start)
a(n) = (1/2)*(1 + (i*sqrt(2))^n * ChebyshevU(n, -3*i/(2*sqrt(2)))).
a(n) = (1/2)*( 1 + Sum_{j=0..floor(n/2)} binomial(n-k,k)*2^k*3^(n-2*k) ). (End)

cp's OEIS Frontend

A193722 Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.

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A193842 Triangular array: the fission of the polynomial sequence ((x+1)^n: n >= 0) by the polynomial sequence ((x+2)^n: n >= 0). (Fission is defined at Comments.)

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A090040 (3*6^n + 2^n)/4.

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A193651 a(n) = ((2*n + 1)!! + 1)/2.

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A193657 First difference of A002627.

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A193664 Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q=Pascal's triangle. (See Comments.)

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