cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A081045 10th binomial transform of (1,9,0,0,0,0,0,...).

Original entry on oeis.org

1, 19, 280, 3700, 46000, 550000, 6400000, 73000000, 820000000, 9100000000, 100000000000, 1090000000000, 11800000000000, 127000000000000, 1360000000000000, 14500000000000000, 154000000000000000, 1630000000000000000, 17200000000000000000, 181000000000000000000
Offset: 0

Views

Author

Paul Barry, Mar 04 2003

Keywords

Comments

From Bernard Schott, Nov 12 2022: (Start)
For n >= 1, a(n-1) is the number of digits 1 (or any nonzero digit) that are necessary to write all the n-digit integers, while the corresponding number of digits 0 to write all these n-digit integers is A212704(n-1) for n >=2.
E.g.: a(2-1) = 19 since 19 digits 2's are required to write integers with a digit 2 from 10 up to 99: {12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92}.
First difference of A053541. (End)

Links

Crossrefs

Cf. A053541, A081044, A081043, A212704.

Programs

  • Magma
    [(9*n+10)*10^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
  • Mathematica
    CoefficientList[Series[(1 - x)/(1 - 10 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{20,-100},{1,19},20] (* Harvey P. Dale, Dec 28 2023 *)

Formula

a(n) = 20*a(n-1) - 100*a(n-2); a(0)=1, a(1)=19.
a(0)=1; for n>= 1, a(n) = (9*n+10)*10^(n-1) = 10^(n-1)*A017173(n+1).
a(n) = Sum_{k=0..n} (k+1)*9^k*binomial(n, k).
G.f.: (1-x)/(1-10*x)^2.
a(n) = A053541(n+1) - A053541(n), for n >= 1. - Bernard Schott, Nov 12 2022
E.g.f.: exp(10*x)*(1 + 9*x). - Stefano Spezia, Jan 31 2025

A081044 9th binomial transform of (1,8,0,0,0,0,0,0,.....).

Original entry on oeis.org

1, 17, 225, 2673, 29889, 321489, 3365793, 34543665, 349156737, 3486784401, 34480423521, 338218086897, 3295011258945, 31914537622353, 307565765227809, 2951106226689969, 28207085096966913, 268687927383516945
Offset: 0

Views

Author

Paul Barry, Mar 04 2003

Keywords

Comments

Also number of (n+1)-digit numbers with exactly one '9' in their decimal expansion. Nine can be replaced by any nonzero digit 1..9. - Zak Seidov, Jul 11 2016

Links

Crossrefs

Cf. A081043, A081045.

Programs

  • Mathematica
    Table[(8n+9)9^(n-1),{n,0,30}] (*or*) LinearRecurrence[{18, -81}, {1, 17}, 40] (* Vincenzo Librandi, Feb 23 2012 *)
  • PARI
    a(n) = (8*n+9)*9^(n-1); \\ Altug Alkan, Jul 18 2016

Formula

a(n) = 18*a(n-1)-81*a(n-2), a(0)=0, a(1)=17.
a(n) = (8n+9)*9^(n-1).
a(n) = Sum_{k=0..n} (k+1)*8^k*binomial(n, k).
G.f.: (1-x)/(1-9x)^2.
E.g.f.: (1 + 8*x)*exp(9*x). - Ilya Gutkovskiy, Jul 18 2016

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

Views

Author

Stefano Spezia, Jan 31 2025

Keywords

Examples

			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, ...
   ...

Crossrefs

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).

Programs

  • Mathematica
    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

Formula

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.
Showing 1-3 of 3 results.

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