A081108 -id:A081108 - cp's OEIS frontend

A081109 9th binomial transform of (1,1,0,0,0,0,0,...).

1, 10, 99, 972, 9477, 91854, 885735, 8503056, 81310473, 774840978, 7360989291, 69735688020, 659002251789, 6213449802582, 58462914051567, 549043018919064, 5147278302366225, 48178524910147866, 450283905890997363

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) + 8*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 (18, -81).

Cf. A081108, A081122, A006234.

[(n+9)*9^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

CoefficientList[Series[(1 - 8 x) / (1 - 9 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)

a(n) = 18*a(n-1) - 81*a(n-2), a(0) = 1, a(1) = 10.
a(n) = (n + 9)*9^(n-1).
G.f.: (1 - 8*x)/(1 - 9*x)^2.
E.g.f.: exp(9*x)*(1 + x). - Stefano Spezia, Mar 04 2023

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

Original entry on oeis.org

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

A081122 10th binomial transform of (1,1,0,0,0,0,...).

Original entry on oeis.org

1, 11, 120, 1300, 14000, 150000, 1600000, 17000000, 180000000, 1900000000, 20000000000, 210000000000, 2200000000000, 23000000000000, 240000000000000, 2500000000000000, 26000000000000000, 270000000000000000

Offset: 0

Paul Barry, Mar 07 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (20, -100).

Cf. A081109, A081108.

[(n+10)*10^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

CoefficientList[Series[(1 - 9 x) / (1 - 10 x)^2, {x, 0, 20}], x] (* Vincenzo Librandi, Aug 06 2013 *)

a(n) = 20*a(n-1) - 100*a(n-2), a(0) = 1, a(1) = 11.
a(n) = (n + 10)*10^(n-1).
G.f.: (1 - 9*x)/(1 - 10*x)^2.
E.g.f.: exp(10*x)*(1 + x). - Stefano Spezia, Mar 04 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)).

cp's OEIS Frontend

A081109 9th binomial transform of (1,1,0,0,0,0,0,...).

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

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A081122 10th 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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