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 1-4 of 4 results.

A104878 A sum-of-powers number triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 15, 13, 5, 1, 1, 6, 31, 40, 21, 6, 1, 1, 7, 63, 121, 85, 31, 7, 1, 1, 8, 127, 364, 341, 156, 43, 8, 1, 1, 9, 255, 1093, 1365, 781, 259, 57, 9, 1, 1, 10, 511, 3280, 5461, 3906, 1555, 400, 73, 10, 1, 1, 11, 1023, 9841, 21845
Offset: 0

Views

Author

Paul Barry, Mar 28 2005

Keywords

Comments

Columns are partial sums of the columns of A004248. Row sums are A104879. Diagonal sums are A104880.
The rows of this triangle (apart from the initial "1" in each row) are the antidiagonals of rectangle A055129. The diagonals of this triangle (apart from the initial "1") are the rows of rectangle A055129. The columns of this triangle (apart from the leftmost column) are the same as the columns of rectangle A055129 but shifted downward. - Mathew Englander, Dec 21 2020

Examples

			Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 15, 13,  5,  1;
  1,  6, 31, 40, 21,  6,  1;
  ...

Crossrefs

Cf. A004248 (first differences by column), A104879 (row sums), A104880 (antidiagonal sums), A125118 (version of this triangle with fewer terms).
This triangle (ignoring the leftmost column) is a rotation of rectangle A055129.
Columns (adjusting offset as necessary): A000012, A000027, A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
T(2n,n) gives A031973.

Programs

  • Maple
    A104878 :=proc(n,k): if k = 0 then 1 elif k=1 then n elif k>=2 then (k^(n-k+1)-1)/(k-1) fi: end: for n from 0 to 7 do seq(A104878(n,k), k=0..n) od; seq(seq(A104878(n,k), k=0..n), n=0..10); # Johannes W. Meijer, Aug 21 2011

Formula

T(n, k) = if(k=1, n, if(k<=n, (k^(n-k+1)-1)/(k-1), 0));
G.f. of column k: x^k/((1-x)(1-k*x)). [corrected by Werner Schulte, Jun 05 2019]
T(n, k) = A069777(n+1,k)/A069777(n,k). [Johannes W. Meijer, Aug 21 2011]
T(n, k) = A055129(n+1-k, k) for n >= k > 0. - Mathew Englander, Dec 19 2020

A104881 Triangle T(n,k) = Sum_{j=0..k} (n-k)^(k-j), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 15, 5, 1, 1, 6, 21, 40, 31, 6, 1, 1, 7, 31, 85, 121, 63, 7, 1, 1, 8, 43, 156, 341, 364, 127, 8, 1, 1, 9, 57, 259, 781, 1365, 1093, 255, 9, 1, 1, 10, 73, 400, 1555, 3906, 5461, 3280, 511, 10, 1, 1, 11, 91, 585, 2801, 9331, 19531, 21845, 9841, 1023, 11, 1
Offset: 0

Views

Author

Paul Barry, Mar 28 2005

Keywords

Comments

Reverse of triangle A104878.

Examples

			Triangle begins as:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  3,  1;
  1, 4,  7,  4, 1;
  1, 5, 13, 15, 5, 1;

Links

Crossrefs

Cf. A104878, A104879 (row sums), A104882 (diagonal sums).

Programs

  • Magma
    [(&+[ (n-k)^(k-j): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 15 2021
  • Mathematica
    T[n_, k_]:= If[k==n, 1, Sum[(n-k)^(k-j), {j,0,k}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 15 2021 *)
  • Sage
    flatten([[sum((n-k)^(k-j) for j in (0..k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 15 2021

Formula

T(n, k) = Sum_{j=0..k} (n-k)^(k-j).
Sum_{k=0..n} T(n, k) = A104879(n).
Sum_{k=0..floor(n/2)} T(k, n-k) = A104882(n).

A104882 Diagonal sums of number triangle A104881.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 24, 45, 85, 170, 351, 749, 1656, 3758, 8776, 21013, 51473, 129018, 329939, 860901, 2288528, 6192526, 17047248, 47693661, 135554549, 391099370, 1144867871, 3398656893, 10226072720, 31173964942, 96240485104, 300777706053
Offset: 0

Views

Author

Paul Barry, Mar 28 2005

Keywords

Links

Crossrefs

Cf. A104879, A104881.

Programs

  • Magma
    [(&+[ (&+[ (n-2*k)^(k-j) : j in [0..k]]) : k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jun 15 2021
  • Mathematica
    Table[Sum[If[j==k, 1, (n-2*k)^(k-j)], {k, 0, Floor[n/2]}, {j,0,k}], {n,0,40}] (* G. C. Greubel, Jun 15 2021 *)
  • Sage
    [sum(sum((n-2*k)^(k-j) for j in (0..k)) for k in (0..n//2)) for n in (0..40)] # G. C. Greubel, Jun 15 2021

Formula

a(n) = Sum_{k=0..floor(n/2)} ( Sum_{j=0..k} (n-2*k)^(k-j) ).
a(n) = Sum_{k=0..floor(n/2)} A104881(n-k, k).

A134195 Antidiagonal sums of square array A126885.

Original entry on oeis.org

1, 3, 7, 15, 32, 72, 178, 494, 1543, 5373, 20581, 85653, 383494, 1833250, 9301792, 49857540, 281193501, 1663183383, 10286884195, 66365330811, 445598473612, 3107611606908, 22470529228910, 168190079241210, 1301213084182483, 10391369994732593, 85553299734530113
Offset: 0

Views

Author

Gary W. Adamson, Oct 12 2007

Keywords

Comments

Conjecture: partial sums of A104879. - Sean A. Irvine, Jul 14 2022

Examples

			a(4) = 1 + 5 + 11 + 10 + 5 = 32.

Crossrefs

Cf. A126885.

Programs

  • Maxima
    T(n, k) := if k = 1 then 1 else n*T(n, k - 1) + k$ /* A126885 */
a(n) := sum(T(n - k + 1, k), k, 1, n + 1)$
makelist(a(n), n, 0, 50); /* Franck Maminirina Ramaharo, Jan 26 2019 */
Showing 1-4 of 4 results.

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