A079028 -id:A079028 - cp's OEIS frontend

A081104 Duplicate of A079028.

1, 5, 24, 112, 512, 2304, 10240, 45056, 196608, 851968, 3670016, 15728640

Offset: 1

dead

A081106 6th binomial transform of (1,1,0,0,0,0,...).

1, 7, 48, 324, 2160, 14256, 93312, 606528, 3919104, 25194240, 161243136, 1027924992, 6530347008, 41358864384, 261213880320, 1645647446016, 10344069660672, 64885527871488, 406239826673664, 2538998916710400, 15843353240272896, 98716277881700352, 614234617930579968

Offset: 0

Paul Barry, Mar 07 2003

Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 5*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 (12,-36).

Cf. A079028, A081105, A006234.

[(n+6)*6^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

CoefficientList[Series[(1 - 5 x)/(1 - 6 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{12,-36},{1,7},30] (* Harvey P. Dale, Nov 07 2013 *)

a(n) = 12*a(n-1) - 36*a(n-2) with n > 1, a(0) = 1, a(1) = 7.
a(n) = (n + 6)*6^(n-1).
G.f.: (1 - 5*x)/(1 - 6*x)^2.
E.g.f.: exp(6*x)*(1 + x). - Stefano Spezia, Mar 05 2023

A082307 Expansion of e.g.f. (1+x)exp(3x)*cosh(x).

Original entry on oeis.org

1, 4, 16, 66, 280, 1208, 5248, 22816, 98944, 427392, 1838080, 7870976, 33568768, 142637056, 604045312, 2550276096, 10737713152, 45097779200, 188979871744, 790276734976, 3298540650496, 13743907405824, 57174629810176

Offset: 0

Paul Barry, Apr 09 2003

Binomial transform of A002306; a(n)=(A082308(n)+A079028(n))/2

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-52,96,-64).

Cf. A082308.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(3*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018

With[{nmax = 50}, CoefficientList[Series[(1 + x)*Exp[3*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 16 2018 *)

x='x+O('x^30); Vec(serlaplace((1+x)*exp(3*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

a(n) = ((n+2)*2^(n-1) + (n+4)*4^(n-1))/2.
G.f.: ((1-3x)/(1-4x)^2 + (1-x)/(1-2x)^2)/2.
E.g.f. (1+x)*exp(3*x)*cosh(x).

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

A235113 Irregular triangle read by rows: T(n,k) = number of independent vertex subsets of size k of the graph g_n obtained by attaching two pendant edges to each vertex of the complete graph K_n (0 <= k <= 2n).

Original entry on oeis.org

1, 1, 3, 1, 1, 6, 10, 6, 1, 1, 9, 27, 38, 27, 9, 1, 1, 12, 52, 116, 150, 116, 52, 12, 1, 1, 15, 85, 260, 490, 602, 490, 260, 85, 15, 1, 1, 18, 126, 490, 1215, 2052, 2436, 2052, 1215, 490, 126, 18, 1, 1, 21, 175, 826, 2541, 5467, 8547, 9900, 8547, 5467, 2541, 826, 175, 21, 1

Offset: 0

Emeric Deutsch, Jan 13 2014

Sum of entries in row n = 2^{2n-4}*(4 + n) = A079028(n).

In the Maple program P[n] gives the independence polynomial of the graph g_n.

			Row 1 is 1,3,1; indeed, K_1 is the one-vertex graph and after attaching two pendant vertices we obtain the path graph ABC; the independent vertex subsets are: empty, {A}, {B}, {C}, and {A, C}.
Triangle begins:
1;
1,3,1;
1,6,10,6,1;
1,9,27,38,27,9,1;
1,12,52,116,150,116,52,12,1;

E. Mandrescu, Unimodality of some independence polynomials via their palindromicity, Australasian J. of Combinatorics, 53, 2012, 77-82.
D. Stevanovic, Graphs with palindromic independence polynomial, Graph Theory Notes of New York, 34, 1998, 31-36.

Cf. A079028

G := (1-z-x*z-x^2*z)/(1-z-2*x*z-x^2*z)^2: Gser := simplify(series(G, z = 0, 10)): for n from 0 to 9 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 9 do seq(coeff(P[n], x, i), i = 0 .. 2*n) end do;# yields sequence in triangular form

Generating polynomial of row n (n>=0) is (1+x)^{2n-2}*((1+x)^2 + nx) (it is palindromic).
Bivariate generating polynomial: G(x,z) = (1-z-xz-x^2*z)/(1-z-2xz-x^2*z)^2.
G(1/x, x^2*z) = G(x,z) (this implies the above mentioned palindromicity).

cp's OEIS Frontend

A081104 Duplicate of A079028.

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A081106 6th binomial transform of (1,1,0,0,0,0,...).

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A082307 Expansion of e.g.f. (1+x)*exp(3*x)*cosh(x).

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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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A235113 Irregular triangle read by rows: T(n,k) = number of independent vertex subsets of size k of the graph g_n obtained by attaching two pendant edges to each vertex of the complete graph K_n (0 <= k <= 2n).

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A082307 Expansion of e.g.f. (1+x)exp(3x)*cosh(x).

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