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.

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A258445 Irregular triangle related to Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 4, 4, 6, 4, 4, 1, 1, 1, 1, 5, 5, 10, 10, 10, 5, 5, 1, 1, 1, 1, 6, 6, 15, 15, 20, 15, 15, 6, 6, 1, 1, 1, 1, 7, 7, 21, 21, 35, 35, 35, 21, 21, 7, 7, 1, 1, 1, 1, 8, 8, 28, 28, 56, 56, 70, 56, 56, 28, 28, 8, 8, 1, 1, 1, 1, 9, 9, 36, 36, 84, 84, 126, 126, 126, 84, 84, 36, 36, 9, 9, 1, 1
Offset: 1

Views

Author

Craig Knecht, May 30 2015

Keywords

Comments

The sequence of row lengths of this irregular triangle T(n, k) is A005408(n-1) = 2*n -1.
This array represents the height of water retention between a collection of cylinders whose height and arrangement are specified by Pascal's triangle.
The row sums for this retention are A164991.
Each term is the minimum of 3 terms of Pascal's triangle: 2 terms below and 1 above when k is odd, and 2 terms above and 1 below when k is even. - Michel Marcus, Jun 11 2015

Examples

			The irregular triangle T(n, k) starts:
n\k 1 2 3 4  5  6  7  8  9 10 11 12 13 14 15 16 17
1:  1
2:  1 1 1
3:  1 1 2 1  1
4:  1 1 3 3  3  1  1
5:  1 1 4 4  6  4  4  1  1
6:  1 1 5 5 10 10 10  5  5  1  1
7:  1 1 6 6 15 15 20 15 15  6  6  1  1
8:  1 1 7 7 21 21 35 35 35 21 21  7  7  1  1
9:  1 1 8 8 28 28 56 56 70 56 56 28 28  8  8  1  1
... Reformatted. - _Wolfdieter Lang_, Jun 26 2015

Links

Crossrefs

Cf. A007318 (Pascal's triangle), A164991.

Programs

  • PARI
    tabf(nn) = {for (n=1, nn, for (k=1, 2*n-1, kk = (k+1)\2; if (k%2, v = min(binomial(n-1, kk-1), min(binomial(n, kk-1), binomial(n, kk))), v = min(binomial(n, kk), min(binomial(n-1, kk-1), binomial(n-1, kk)))); print1(v, ", ");); print(););} \\ Michel Marcus, Jun 16 2015

Formula

T(n, 2*m) = Min(P(n-1, m-1), P(n-1, m), P(n, m)) with P(n, k) = A007318(n, k) = binomial(n, k), for m = 1, 2, ..., n-1, and
T(n, 2*m-1) = Min(P(n-1, m-1), P(n, m-1), P(n, m)) for m = 1, 2, ..., n. See the program by Michel Marcus. - Wolfdieter Lang, Jun 27 2015

A303602 a(n) = Sum_{k = 0..n} k*binomial(2*n+1, k).

Original entry on oeis.org

0, 3, 25, 154, 837, 4246, 20618, 97140, 447661, 2028478, 9070110, 40122028, 175913250, 765561564, 3310623412, 14238676712, 60949133949, 259809601870, 1103420316566, 4670886541308, 19714134528598, 82985455688276, 348481959315660, 1460179866076504, 6106070639175122
Offset: 0

Views

Author

Bruno Berselli, May 09 2018

Keywords

Comments

Second bisection of A185251; the first bisection is A002699.
The terms are not congruent to 5 (mod 6).

Crossrefs

Cf. A000346, A002699, A005408, A047226, A164991, A185251.

Programs

  • Maple
    seq(add(k*binomial(2*n+1,k),k=0..n),n=0..24); # Paolo P. Lava, May 10 2018
  • Mathematica
    Table[Sum[k Binomial[2 n + 1, k], {k, 0, n}], {n, 0, 30}]
CoefficientList[Series[(1 + 4*x - Sqrt[1 - 4*x]) / (2*(1 - 4*x)^2), {x, 0, 25}], x] (* Vaclav Kotesovec, May 10 2018 *)
  • PARI
    a(n)=(2*n+1)*(4^n-binomial(2*n,n))/2 \\ Charles R Greathouse IV, Oct 23 2023
  • Sage
    [(2*n+1)*(4^n-binomial(2*n,n))/2 for n in (0..30)]

Formula

E.g.f.: ((1 + 8*x)*exp(2*x) - (1 + 4*x)*I_0(2*x) - 4*x*I_1(2*x))*exp(2*x)/2, where I_m(.) is the modified Bessel function of the first kind.
From Vaclav Kotesovec, May 10 2018: (Start)
G.f.: (1 + 4*x - sqrt(1 - 4*x)) / (2*(1 - 4*x)^2).
D-finite with recurrence: n*(2*n-1)*a(n) = 2*(2*n+1)*(4*n-3)*a(n-1) - 8*(2*n-1)*(2*n+1)*a(n-2). (End)
a(n) = (2*n + 1)*(4^n - binomial(2*n, n))/2.
a(n+1) - 4*a(n) = A164991(2*n+3).
Showing 1-2 of 2 results.

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