A081042 -id:A081042 - cp's OEIS frontend

A081041 6th binomial transform of (1,5,0,0,0,0,0,0,...).

1, 11, 96, 756, 5616, 40176, 279936, 1912896, 12877056, 85660416, 564350976, 3688436736, 23944605696, 154551545856, 992612745216, 6347497291776, 40435908673536, 256721001578496, 1624959306694656, 10257555623510016

Offset: 0

Paul Barry, Mar 04 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.
Index entries for linear recurrences with constant coefficients, signature (12,-36).

Cf. A081040, A081042.

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

CoefficientList[Series[(1 - x)/(1 - 6 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{12,-36},{1,11},20] (* Harvey P. Dale, Mar 04 2019 *)

a(n) = 12*a(n-1) - 36*a(n-2) for n>1, a(0)=1, a(1)=9.
a(n) = (5*n+6)*6^(n-1).
a(n) = Sum_{k=0..n} (k+1)*5^k*binomial(n, k).
G.f.: (1-x)/(1-6*x)^2.
E.g.f.: exp(6*x)*(1 + 5*x). - Stefano Spezia, Jan 31 2025

A081043 8th binomial transform of (1,7,0,0,0,0,0,...).

Original entry on oeis.org

1, 15, 176, 1856, 18432, 176128, 1638400, 14942208, 134217728, 1191182336, 10468982784, 91268055040, 790273982464, 6803228196864, 58274116272128, 496979255754752, 4222124650659840, 35747322042253312, 301741175033823232

Offset: 0

Paul Barry, Mar 04 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A081042, A081044.

I:=[1, 15]; [n le 2 select I[n] else 16*Self(n-1)-64*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 23 2012

LinearRecurrence[{16,-64},{1,15},20] (* or *) Table[(7n+8)8^(n-1),{n,0,20}] (* Harvey P. Dale, Feb 22 2012 *)

a(n) = 16*a(n-1) - 64*a(n-2), a(0)=1, a(1)=15.
a(n) = (7n+8)*8^(n-1).
a(n) = Sum_{k=0..n} (k+1)*7^k*binomial(n, k).
G.f.: (1-x)/(1-8*x)^2.
E.g.f.: exp(8*x)*(1 + 7*x). - Stefano Spezia, Jan 31 2025

A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

Original entry on oeis.org

1, -1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 5, 1, 0, 1, 20, 21, 7, 1, 0, 1, 48, 81, 40, 9, 1, 0, 1, 112, 297, 208, 65, 11, 1, 0, 1, 256, 1053, 1024, 425, 96, 13, 1, 0, 1, 576, 3645, 4864, 2625, 756, 133, 15, 1, 0, 1, 1280, 12393, 22528, 15625, 5616, 1225, 176, 17, 1

Offset: 0

Stefano Spezia, Jan 31 2025

			The array begins as:
   1, 1,   1,    1,     1,     1, ...
  -1, 1,   3,    5,     7,     9, ...
   0, 1,   8,   21,    40,    65, ...
   0, 1,  20,   81,   208,   425, ...
   0, 1,  48,  297,  1024,  2625, ...
   0, 1, 112, 1053,  4864, 15625, ...
   0, 1, 256, 3645, 22528, 90625, ...
   ...

Cf. A000012 (k=1 or n=0), A000567 (n=2), A001792 (k=2), A007778, A060747 (n=1), A081038 (k=3), A081039 (k=4), A081040 (k=5), A081041 (k=6), A081042 (k=7), A081043 (k=8), A081044 (k=9), A081045 (k=10), A103532, A154955, A380748 (antidiagonal sums).

A[0,0]:=1; A[1,0]:=-1; A[n_,k_]:=((k-1)*n+k)k^(n-1); Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1-x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+(k-1)*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n,k) = ((k - 1)*n + k)*k^(n-1) with A(0,0) = 1.
A(n,k) = n! * [x^n] exp(k*x)*(1 + (k - 1)*x).
A(n,0) = A154955(n+1).
A(3,n) = A103532(n-1) for n > 0.
A(n,n) = A007778(n) for n > 0.

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A081041 6th binomial transform of (1,5,0,0,0,0,0,0,...).

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A081043 8th binomial transform of (1,7,0,0,0,0,0,...).

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A380747 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.

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