A081106 -id:A081106 - cp's OEIS frontend

A081107 7th binomial transform of (1,1,0,0,0,0,...).

1, 8, 63, 490, 3773, 28812, 218491, 1647086, 12353145, 92236816, 686011319, 5084554482, 37569208117, 276825744020, 2034669218547, 14920907602678, 109193914728689, 797590333670424, 5815762849680175, 42338753545671674, 307770170005074861, 2234183456333136028

Offset: 0

Paul Barry, Mar 07 2003

Main diagonal of array defined by m(0,j) = j; m(i,0) = i and m(i,j) = m(i-1,j) + 6*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A081106, A081108, A006234.

[(n+7)*7^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

CoefficientList[Series[(1 - 6 x)/(1 - 7 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)

a(n) = 14*a(n-1) - 49*a(n-2) with n > 1, a(0) = 1, a(1) = 8.
a(n) = (n + 7)*7^(n-1).
G.f.: (1 - 6*x)/(1 - 7*x)^2.
E.g.f.: exp(7*x)*(1 + x). - Stefano Spezia, Mar 05 2023

A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 8, 4, 1, 5, 20, 15, 5, 1, 6, 48, 54, 24, 6, 1, 7, 112, 189, 112, 35, 7, 1, 8, 256, 648, 512, 200, 48, 8, 1, 9, 576, 2187, 2304, 1125, 324, 63, 9, 1, 10, 1280, 7290, 10240, 6250, 2160, 490, 80, 10, 1, 11, 2816, 24057, 45056, 34375, 14256, 3773, 704, 99, 11, 1

Offset: 0

Paul D. Hanna, Nov 23 2003

The main diagonal is A089945: {T(n,n)=(2*n+1)*(n+1)^(n-1), n>=0}; the hyperbinomial transform of the main diagonal is the next lower diagonal in the array (A089946): {T(n+1,n) = 2*(n+1)*(n+2)^(n-1), n>=0}.

			Rows begin:
  {1, 2, 3, 4, 5, 6, 7,..},
  {1, 3, 8, 20, 48, 112, 256,..},
  {1, 4, 15, 54, 189, 648, 2187,..},
  {1, 5, 24, 112, 512, 2304, 10240,..},
  {1, 6, 35, 200, 1125, 6250, 34375,..},
  {1, 7, 48, 324, 2160, 14256, 93312,..},
  {1, 8, 63, 490, 3773, 28812, 218491,..},..

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals).

Cf. A089945, A089946.
Rows : A000027, A001792, A006234, A079028, A081105, A081106, A081107, A081108, A081109, A081122.
Columns : A000012, A000027, A005563.

A089944[n_, k_] := (k + n + 1)*(n + 1)^(k - 1);
Table[A089944[k, n - k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2025 *)

T(n,k)=if(n<0 || k<0,0,(k+n+1)*(n+1)^(k-1))

T(n,k) = (k+n+1)*(n+1)^(k-1).

E.g.f.: (1+x)*exp(x)/(1-y*exp(x)).

A082309 Expansion of e.g.f.: (1+x)exp(5x)*cosh(x).

Original entry on oeis.org

1, 6, 36, 218, 1336, 8280, 51776, 325792, 2057856, 13023104, 82456576, 521826816, 3298727936, 20822038528, 131210919936, 825373859840, 5182772248576, 32487861092352, 203308891897856, 1270289732337664, 7924975155019776

Offset: 0

Paul Barry, Apr 09 2003

Binomial transform of A082307.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (20,-148,480,-576).

Cf. A082308, A082307, A082306, A082305.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(5*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018

With[{nn=30},CoefficientList[Series[(1+x)Exp[5x]Cosh[x],{x,0,nn}],x]Range[0,nn]!] (* or *) LinearRecurrence[{20,-148,480,-576},{1,6,36,218},30] (* Harvey P. Dale, Aug 27 2012 *)

x='x+O('x^30); Vec(serlaplace((1+x)*exp(5*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

a(n) = (A081106(n) + A079028(n))/2.
a(n) = ((n+4)*4^(n-1) + (n+6)*6^(n-1))/2.
G.f.: ((1-5*x)/(1-6*x)^2 + (1-3*x)/(1-4*x)^2)/2.
From Harvey P. Dale, Aug 27 2012: (Start)
E.g.f.: (1+x)*exp(5*x)*cosh(x).
a(n) = 20*a(n-1) - 148*a(n-2) + 480*a(n-3) - 576*a(n-4), n>3. (End)

Definition clarified by Harvey P. Dale, Aug 27 2012

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)x - c)B(x, n-1) - abB(x, n-2) with a = 3, b = 2, and c = 0.

Original entry on oeis.org

1, 0, 1, -6, 0, 6, 0, -42, 0, 36, 36, 0, -288, 0, 216, 0, 468, 0, -1944, 0, 1296, -216, 0, 4536, 0, -12960, 0, 7776, 0, -4104, 0, 38880, 0, -85536, 0, 46656, 1296, 0, -51840, 0, 311040, 0, -559872, 0, 279936, 0, 32400, 0, -544320, 0, 2379456, 0, -3639168, 0, 1679616

Offset: 0

Roger L. Bagula, Mar 23 2008

			Triangle begins as:
     1;
     0,     1;
    -6,     0,      6;
     0,   -42,      0,    36;
    36,     0,   -288,     0,    216;
     0,   468,      0, -1944,      0,   1296;
  -216,     0,   4536,     0, -12960,      0,    7776;
     0, -4104,      0, 38880,      0, -85536,       0, 46656;
  1296,     0, -51840,     0, 311040,      0, -559872,     0, 279936;

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 93

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

Cf. A000567, A002414, A002419, A016921, A027810, A030192, A034265, A051843, A054487, A055848, A081106, A136531.

f:= func< n,k | k eq 0 select (-1)^Floor(n/2) else (-1)^Floor((n-k)/2)*6^Floor((k-1)/2)*(1/k)*(6*Floor((n-k)/2) +k)*Binomial(Floor((n-k)/2) +k-1, k-1) >;
A136526:= func< n,k | ((n+k+1) mod 2)*6^Floor(n/2)*f(n,k) >;
[A136526(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 22 2022

(* First program *)
a= (b+1)/(b-1); c=0; b=2;
B[x_, n_]:= B[x, n]= If[n<2, x^n, ((1+a+b)*x -c)*B[x, n-1] -a*b*B[x, n-2]];
Table[CoefficientList[B[x,n], x], {n,0,10}]//Flatten
(* Second program *)
B[x_, n_]:= 6^(n/2)*(ChebyshevU[n, Sqrt[3/2]*x] -(5*x/Sqrt[6])*ChebyshevU[n-1, Sqrt[3/2]*x]);
Table[CoefficientList[B[x, n], x]/6^Floor[n/2], {n,0,16}]//Flatten (* G. C. Greubel, Sep 22 2022 *)

def f(n,k):
    if (k==0): return (-1)^(n//2)
    else: return (-1)^((n-k)//2)*6^((k-1)//2)*(1/k)*(6*((n-k)//2) + k)*binomial(((n-k)//2) +k-1, k-1)
def A136526(n,k): return ((n+k+1)%2)*6^(n//2)*f(n,k)
flatten([[A136526(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 22 2022

T(n, k) = coefficients of the polynomials defined by B(x, n) = ((1 + a + b)*x - c)*B(x, n - 1) - a*b*B(x, n - 2) with B(x, 0) = 1, B(x, 1) = x, a = 3, b = 2, and c = 0.
From G. C. Greubel, Sep 22 2022: (Start)
T(n, k) = coefficients of the polynomials defined by B(x, n) = 6^(n/2)*(ChebyshevU(n, sqrt(3/2)*x) - (5*x/sqrt(6))*ChebyshevU(n-1, sqrt(3/2)*x)).
T(n, k) = (1/2)*(1+(-1)^(n+k))*6^floor(n/2)*f(n, k), where f(n, k) = (-1)^floor((n -k)/2)*6^floor((k-1)/2)*(1/k)*(6*floor((n-k)/2) + k)*binomial(floor((n-k)/2) + k -1, k-1), for k >= 1, and (-1)^floor(n/2) for k = 0.
T(n, 0) = (1/2)*(1+(-1)^n)*(-6)^floor(n/2).
T(n, 1) = (1/2)*(1-(-1)^n)*(-6)^floor((n-1)/2)*A016921(floor((n-1)/2)), n >= 1.
T(n, 2) = (1/2)*(1+(-1)^n)*(-1)^(1+Floor((n+1)/2))*6^floor((n+1)/2)*A000567(floor( (n+1)/2)), n >= 2.
T(n, 3) = (1/2)*(1-(-1)^n)*(-6)^floor((n+1)/2)*A002414(floor((n-1)/2)), n >= 3.
T(n, 4) = (3/2)*(1+(-1)^n)*(-6)^floor((n+1)/2)*A002419(floor((n-1)/2)), n >= 4.
T(n, 5) = 18*(1-(-1)^n)*(-6)^floor((n-1)/2)*A051843(floor((n-3)/2)), n >= 5.
T(n, n) = 6^(n-1) + (5/6)*[n=0].
T(n, n-2) = -6*A081106(n-2), n >= 2.
Sum_{k=0..n} T(n, k) = -6*A030192(n-3), n>= 0.
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - 5*[n=2].
G.f.: (1 - 5*x*y)/(1 - 6*x*y + 6*y^2). (End)

Edited by G. C. Greubel, Sep 22 2022

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A081107 7th binomial transform of (1,1,0,0,0,0,...).

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A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

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A082309 Expansion of e.g.f.: (1+x)*exp(5*x)*cosh(x).

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A136526 Coefficients polynomials B(x, n) = ((1 + a + b)*x - c)*B(x, n-1) - a*b*B(x, n-2) with a = 3, b = 2, and c = 0.

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A082309 Expansion of e.g.f.: (1+x)exp(5x)*cosh(x).

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)x - c)B(x, n-1) - abB(x, n-2) with a = 3, b = 2, and c = 0.