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.

A101494 Triangle, read by rows, where T(n,k) = Sum_{j=0..n-k-1} C(j+k,j)*T(n-1,j+k) for n>k>=0 with T(n,n)=1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 8, 4, 1, 1, 23, 23, 13, 5, 1, 1, 66, 73, 44, 19, 6, 1, 1, 210, 253, 162, 73, 26, 7, 1, 1, 733, 948, 643, 302, 111, 34, 8, 1, 1, 2781, 3817, 2724, 1337, 506, 159, 43, 9, 1, 1, 11378, 16433, 12259, 6266, 2457, 788, 218, 53, 10, 1, 1, 49864, 75295
Offset: 0

Views

Author

Paul D. Hanna, Jan 21 2005

Keywords

Comments

Column 0 equals row sums (A026898) shift right.
T(n,k) is the number of m-tuples of nonnegative integers satisfying these two criteria: (i) there are exactly k 0’s, and (ii) the remaining m-k elements are positive integers less than or equal to n-m. - Mathew Englander, Feb 25 2021

Examples

			4th row sum = 23 = (5-0)^0+(5-1)^1+(5-2)^2+(5-3)^3+(5-4)^4.
5th row sum = 66 = (6-0)^0+(6-1)^1+(6-2)^2+(6-3)^3+(6-4)^4+(6-5)^5.
T(6,0) = 66 = 1*23 + 1*23 + 1*13 + 1*5 + 1*1 + 1*1.
T(6,1) = 73 = 1*23 + 2*13 + 3*5 + 4*1 + 5*1.
T(6,2) = 44 = 1*13 + 3*5 + 6*1 + 10*1.
Rows begin:
1;
1, 1;
2, 1, 1;
4, 3, 1, 1;
9, 8, 4, 1, 1;
23, 23, 13, 5, 1, 1;
66, 73, 44, 19, 6, 1, 1;
210, 253, 162, 73, 26, 7, 1, 1;
733, 948, 643, 302, 111, 34, 8, 1, 1;
2781, 3817, 2724, 1337, 506, 159, 43, 9, 1, 1;
11378, 16433, 12259, 6266, 2457, 788, 218, 53, 10, 1, 1;
49864, 75295, 58423, 30953, 12558, 4147, 1163, 289, 64, 11, 1, 1;
232769, 365600, 293902, 160823, 67259, 22878, 6574, 1647, 373, 76, 12, 1, 1; ...

Links

Crossrefs

Cf. A101495, A026898, A089246 (first differences by column), A304357 (antidiagonal sums, empirically), A034856 (fourth diagonal).

Programs

  • GAP
    Flat(List([0..10],n->List([0..n],k->Sum([0..n-k],j->Binomial(j+k,j)*(n-k-j)^j)))); # Muniru A Asiru, Mar 07 2019
  • PARI
    T(n,k)=if(n
  • PARI
    T(n,k)=polcoeff(sum(m=0,n-k, x^m/(1-m*x +x*O(x^(n-k)))^(k+1)),n-k)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Mar 06 2013

Formula

T(n,0) = A026898(n-1).
T(n,k) = Sum_{j=0..n-k} binomial(j+k,j)*(n-k-j)^j. - Vladeta Jovovic, Sep 07 2006
G.f.: A(x,y) = Sum_{n>=0} Sum_{k>=0} x^(n+k)*y^k / (1 - n*x)^(k+1). - Paul D. Hanna, Mar 06 2013
From Mathew Englander, Feb 25 2021: (Start)
G.f. of row n: Sum_{i=0..n} (x+n-i)^i.
T(n,k) = Sum_{j=k..n} A089246(j,k).
Antidiagonal sums: Sum_{j = 0..n} Sum_{i = j..floor((n+j)/2)} binomial(i,j)*(n+j-2*i)^j. (End)

A062807 a(n) = Sum_{i=1..n} i*(n-i)^i.

Original entry on oeis.org

0, 1, 4, 14, 50, 187, 738, 3084, 13652, 63917, 315736, 1641314, 8956110, 51175799, 305527878, 1901829488, 12319405728, 82896050937, 578485474092, 4180313933206, 31237475311690, 241063266361235, 1918899090047882
Offset: 1

Views

Author

Olivier Gérard, Jun 23 2001

Keywords

Links

Crossrefs

Cf. A101495.

Programs

  • Mathematica
    Table[Sum[i (n-i)^i,{i,n}],{n,30}] (* Harvey P. Dale, Dec 25 2017 *)
  • PARI
    a(n)={sum(i=1, n, i*(n - i)^i)} \\ Harry J. Smith, Aug 11 2009

Formula

G.f.: x*Sum_{k>=1} k*x^k/(1 - k*x)^2. - Ilya Gutkovskiy, Oct 11 2018

Extensions

Prior Mathematica program replaced by Harvey P. Dale, Dec 25 2017

A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 12, 6, 1, 0, 0, 5, 32, 27, 8, 1, 0, 0, 6, 80, 108, 48, 10, 1, 0, 0, 7, 192, 405, 256, 75, 12, 1, 0, 0, 8, 448, 1458, 1280, 500, 108, 14, 1, 0, 0, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 0, 0, 10, 2304
Offset: 0

Views

Author

Franck Maminirina Ramaharo, Oct 14 2018

Keywords

Comments

T(n,k) is the number of length n*k binary words of n consecutive blocks of length k, respectively, one of the blocks having exactly k letters 1, and the other having exactly one letter 0. First column follows from the next definition.
In Kauffman's language, T(n,k) is the total number of Jordan trails that are obtained by placing state markers at the crossings of the Pretzel universe P(k, k, ..., k) having n tangles, of k half-twists respectively. In other words, T(n,k) is the number of ways of splitting the crossings of the Pretzel knot shadow P(k, k, ..., k) such that the final diagram is a single Jordan curve. The aforementionned binary words encode these operations by assigning each tangle a length k binary words with the adequate choice for splitting the crossings.
Columns are linear recurrence sequences with signature (2*k, -k^2).

Examples

			Square array begins:
    0, 0,   0,    0,     0,      0,      0,      0, ...
    1, 1,   1,    1,     1,      1,      1,      1, ...
    0, 2,   4,    6,     8,     10,     12,     14, ... A005843
    0, 3,  12,   27,    48,     75,    108,    147, ... A033428
    0, 4,  32,  108,   256,    500,    864,   1372, ... A033430
    0, 5,  80,  405,  1280,   3125,   6480,  12005, ... A269792
    0, 6, 192, 1458,  6144,  18750,  46656, 100842, ...
    0, 7, 448, 5103, 28672, 109375, 326592, 823543, ...
    ...
T(3,2) = 3*2^(3 - 1) = 12. The corresponding binary words are 110101, 110110, 111001, 111010, 011101, 011110, 101101, 101110, 010111, 011011, 100111, 101011.

References

  • Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Links

Crossrefs

Antidiagonal sums: A101495.
Column 1 is column 2 of A300453.
Column 2 is column 1 of A300184.
Cf. A104002, A320530.

Programs

  • Mathematica
    T[n_, k_] = If [k > 0, n*k^(n - 1), If[k == 0 && n == 1, 1, 0]];
Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 12}]//Flatten
  • Maxima
    T(n, k) := if k > 0 then n*k^(n - 1) else if k = 0 and n = 1 then 1 else 0$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, nn))$

Formula

T(n,k) = (2*k)*T(n-1,k) - (k^2)*T(n-2,k).
G.f. for columns: x/(1 - k*x)^2.
E.g.f. for columns: x*exp(k*x).
T(n,1) = A001477(n).
T(n,2) = A001787(n).
T(n,3) = A027471(n+1).
T(n,4) = A002697(n).
T(n,5) = A053464(n).
T(n,6) = A053469(n), n > 0.
T(n,7) = A027473(n), n > 0.
T(n,8) = A053539(n).
T(n,9) = A053540(n), n > 0.
T(n,10) = A053541(n), n > 0.
T(n,11) = A081127(n).
T(n,12) = A081128(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.