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.

A125055 Diagonal of symmetric triangle A125053 located immediately below the central terms (A125054).

Original entry on oeis.org

1, 15, 285, 8475, 378105, 23823015, 2018820885, 221605991475, 30596648805105, 5189967817758015, 1061021392126671885, 257296819626005894475, 73023341368629447792105, 23978466652359211809453015
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2006

Keywords

Comments

Triangle A125053 is a variant of triangle A008301 (enumeration of binary trees) such that the leftmost column gives the secant numbers (A000364).

Crossrefs

Cf. A125053, A125054.

Formula

a(n) = A125054(n+1) - 2*A125054(n). - Philippe Deléham, Jul 22 2007

A125053 Variant of triangle A008301, read by rows of 2*n+1 terms, such that the first column is the secant numbers (A000364).

Original entry on oeis.org

1, 1, 3, 1, 5, 15, 21, 15, 5, 61, 183, 285, 327, 285, 183, 61, 1385, 4155, 6681, 8475, 9129, 8475, 6681, 4155, 1385, 50521, 151563, 247065, 325947, 378105, 396363, 378105, 325947, 247065, 151563, 50521, 2702765, 8108295, 13311741, 17908935
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2006, Dec 20 2006

Keywords

Comments

Foata and Han refer to this as the triangle of Poupard numbers h_n(k). - N. J. A. Sloane, Feb 17 2014
Central terms (A125054) equal the binomial transform of the tangent numbers (A000182).

Examples

			If we write the triangle like this:
......................... ...1;
................... ...1, ...3, ...1;
............. ...5, ..15, ..21, ..15, ...5;
....... ..61, .183, .285, .327, .285, .183, ..61;
. 1385, 4155, 6681, 8475, 9129, 8475, 6681, 4155, 1385;
then the first nonzero term is the sum of the previous row:
1385 = 61 + 183 + 285 + 327 + 285 + 183 + 61,
the next term is 3 times the first:
4155 = 3*1385,
and the remaining terms in each row are obtained by the rule illustrated by:
6681 = 2*4155 - 1385 - 4*61;
8475 = 2*6681 - 4155 - 4*183;
9129 = 2*8475 - 6681 - 4*285;
8475 = 2*9129 - 8475 - 4*327;
6681 = 2*8475 - 9129 - 4*285;
4155 = 2*6681 - 8475 - 4*183;
1385 = 2*4155 - 6681 - 4*61.
An alternate recurrence is illustrated by:
4155 = 1385 + 2*(61 + 183 + 285 + 327 + 285 + 183 + 61);
6681 = 4155 + 2*(183 + 285 + 327 + 285 + 183);
8475 = 6681 + 2*(285 + 327 + 285);
9129 = 8475 + 2*(327);
and then for k>n, T(n,k) = T(n,2*n-k).

Links

Crossrefs

Cf. A008301, A000364 (secant numbers, which are the row sums), A125054 (central terms), A125055, A000182, A008282.
Cf. A210111 (left half).

Programs

  • Haskell
    a125053 n k = a125053_tabf !! n !! k
a125053_row n = a125053_tabf !! n
a125053_tabf = iterate f [1] where
f zs = zs' ++ reverse (init zs') where
zs' = (sum zs) : g (map (* 2) zs) (sum zs)
g [x] y = [x + y]
g xs y = y' : g (tail $ init xs) y' where y' = sum xs + y
-- Reinhard Zumkeller, Mar 17 2012
  • Maple
    T := proc(n, k) option remember; local j;
  if n = 1 then 1
elif k = 1 then add(T(n-1, j), j=1..2*n-3)
elif k = 2 then 3*T(n, 1)
elif k > n then T(n, 2*n-k)
else 2*T(n, k-1) - T(n, k-2) - 4*T(n-1, k-2)
  fi end:
seq(print(seq(T(n,k), k=1..2*n-1)), n=1..5); # Peter Luschny, May 11 2014
  • Mathematica
    t[n_, k_] := t[n, k] = If[2*n < k || k < 0, 0, If[n == 0 && k == 0, 1, If[k == 0, Sum[t[n-1, j], {j, 0, 2*n-2}], If[k <= n, t[n, k-1] + 2*Sum[t[n-1, j], {j, k-1, 2*n-1-k}], t[n, 2*n-k]]]]]; Table[t[n, k], {n, 0, 6}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Dec 06 2012, translated from Pari *)
  • PARI
    T(n,k)=if(2*n

Formula

Sum_{k=0..2n} C(2n,k)*T(n,k) = 4^n * A000182(n), where A000182 are the tangent numbers.
Sum_{k=0..2n} (-1)^n*C(2n,k)*T(n,k) = (-4)^n.

A130847 Triangle T(n,k), 0<=k<=n, read by rows, given by [1,2,3,4,5,6,7,8,9,10,...] DELTA [1,1,6,6,15,15,28,28,...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

1, 1, 1, 3, 5, 2, 15, 42, 37, 10, 105, 450, 699, 458, 104, 945, 5775, 13845, 16065, 8866, 1816, 10395, 85995, 293265, 522345, 506028, 248660, 47312
Offset: 0

Views

Author

Philippe Deléham, Jul 21 2007, Jul 25 2007

Keywords

Examples

			Triangle begins:
1;
1, 1;
3, 5, 2;
15, 42, 37, 10;
105, 450, 699, 458, 104;
945, 5775, 13845, 16065, 8866, 1816;
10395, 85995, 293265, 522345, 506028, 248660, 47312 ;...

Formula

Sum_{k, 0<=k<=n}T(n,k)*x^k = A033999(n), A000007(n), A001147(n), A005799(n+1), A125054(n), A126152(n), A126157(n) for x= -2, -1, 0, 1, 2, 3, 4 respectively . T(n,n)=A005799(n).

A210111 Left half of triangle A125053.

Original entry on oeis.org

1, 1, 3, 5, 15, 21, 61, 183, 285, 327, 1385, 4155, 6681, 8475, 9129, 50521, 151563, 247065, 325947, 378105, 396363, 2702765, 8108295, 13311741, 17908935, 21517869, 23823015, 24615741, 199360981, 598082943, 985993845, 1341471567, 1643702325, 1874297343
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 17 2012

Keywords

Links

Crossrefs

Cf. A000364 (left edge), A125054 (right edge); A210108.

Programs

  • Haskell
    a210111 n k = a210111_tabl !! n !! k
a210111_row n = a210111_tabl !! n
a210111_tabl = zipWith take [1..] a125053_tabf

Formula

T(n,k) = A125053(n,k), 0 <= k <= n.
Showing 1-4 of 4 results.

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