A081044 -id:A081044 - cp's OEIS frontend

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

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

Offset: 0

Paul Barry, Mar 04 2003

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)

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A053541, A081044, A081043, A212704.

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

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 *)

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

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.

A380860 Triangle read by rows: T(n,m) (0<=m<=n) = number of positive n-digit numbers that have exactly m copies of a specific, previously selected positive base-10 digit among its digits.

Original entry on oeis.org

1, 8, 1, 72, 17, 1, 648, 225, 26, 1, 5832, 2673, 459, 35, 1, 52488, 29889, 6804, 774, 44, 1, 472392, 321489, 91125, 13770, 1170, 53, 1, 4251528, 3365793, 1141614, 215055, 24300, 1647, 62, 1, 38263752, 34543665, 13640319, 3077109, 433755, 39123, 2205, 71, 1, 344373768, 349156737, 157306536, 41334300, 6980904, 785862, 58968, 2844, 80, 1

Offset: 0

Peter Starek, Feb 06 2025

			Rows n=0..7 of the triangle are:
        1;
        8,       1;
       72,      17,       1;
      648,     225,      26,      1;
     5832,    2673,     459,     35,     1;
    52488,   29889,    6804,    774,    44,    1;
   472392,  321489,   91125,  13770,  1170,   53,  1;
  4251528, 3365793, 1141614, 215055, 24300, 1647, 62, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Row sums give A052268 (for n>=1).

Columns k=0-1 give: A055275, A081044(n-1) (for n>=1).

seq(lprint(seq(floor(9^(n-m)*(binomial(n-1, m-1)+(8/9)*binomial(n-1, m))), m=0..n)), n=0..7);

A380860[n_, m_] := Floor[9^(n-m)*(Binomial[n-1, m-1] + 8/9*Binomial[n-1, m])];
Table[A380860[n, m], {n, 0, 10}, {m, 0, n}] (* Paolo Xausa, Feb 07 2025 *)

T(n,m) = floor(9^(n-m)*(binomial(n-1,m-1)+8/9*binomial(n-1,m))), considering binomial(k,-1)=0 for k>=0 and binomial(k,l)=0 for k>=0 with k

Showing 1-4 of 4 results.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A380860 Triangle read by rows: T(n,m) (0<=m<=n) = number of positive n-digit numbers that have exactly m copies of a specific, previously selected positive base-10 digit among its digits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula