A014301 -id:A014301 - cp's OEIS frontend

A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

1, 1, 0, 1, 1, 2, 1, 2, 4, 6, 1, 3, 7, 13, 22, 1, 4, 11, 24, 46, 80, 1, 5, 16, 40, 86, 166, 296, 1, 6, 22, 62, 148, 314, 610, 1106, 1, 7, 29, 91, 239, 553, 1163, 2269, 4166, 1, 8, 37, 128, 367, 920, 2083, 4352, 8518, 15792, 1, 9, 46, 174, 541, 1461, 3544, 7896

Offset: 0

Peter Luschny, Nov 14 2012

Notation: If a sequence id is starred then the offset and/or some terms are different. Starred terms indicate the variance.
Row sums: [A026641 ] [1,  1,  4,  13,  46,  166,   610]
--
T(j+2, 2) [A000124*] [1*, 2 , 4,   7,  11,  16,     22]
T(j+3, 3) [A003600*] [1*, 2*, 6,  13,  24,  40,     62]
--
T(j  , j) [A072547 ] [1,  0,  2,   6,  22,  80,    296]
T(j+1, j) [A026641 ] [1,  1,  4,  13,  46,  166,   610]
T(j+2, j) [A014300 ] [1,  2,  7,  24,  86,  314,  1163]
T(j+3, j) [A014301*] [1,  3, 11,  40, 148,  553,  2083]
T(j+4, j) [A172025 ] [1,  4, 16,  62, 239,  920,  3544]
T(j+5, j) [A172061 ] [1,  5, 22,  91, 367, 1461,  5776]
T(j+6, j) [A172062 ] [1,  6, 29, 128, 541, 2232,  9076]
T(j+7, j) [A172063 ] [1,  7, 37, 174, 771, 3300, 13820]
--
T(2j  ,j) [Central ] [1,  1,  7,  40, 239, 1461,  9076]
T(2j+1,j) [A183160 ] [1,  2, 11,  62, 367, 2232, 13820]
T(2j+2,j) [        ] [1,  3, 16,  91, 541, 3300, 20476]
T(2j+3,j) [A199033*] [1,  4, 22, 128, 771, 4744, 29618]

			Triangle begins as:
[n]|k->
[0] 1
[1] 1, 0
[2] 1, 1,  2
[3] 1, 2,  4,  6
[4] 1, 3,  7, 13,  22
[5] 1, 4, 11, 24,  46,  80
[6] 1, 5, 16, 40,  86, 166, 296
[7] 1, 6, 22, 62, 148, 314, 610, 1106.

G. C. Greubel, Rows n=0..100 of triangle, flattened

A201635 := proc(n,k) option remember; local j;
if n=k then (-1)^n*add(binomial(-n,j), j=0..n)
else add(A201635(n-1,j), j=0..k) fi end:
for n from 0 to 7 do seq(A(n,k), k=0..n) od;

T[n_, k_]:= T[n, k]= If[k==n, (-1)^n*Sum[Binomial[-n, j], {j, 0, n}], Sum[T[n-1, j], {j, 0, k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 27 2019 *)

{T(n,k) = if(k==n, (-1)^n*sum(j=0,n, binomial(-n,j)), sum(j=0,k, T(n-1,j)))};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 27 2019

@CachedFunction
def A201635(n, k):
    if n==k: return (-1)^n*add(binomial(-n, j) for j in (0..n))
    return add(A201635(n-1, j) for j in (0..k))
for n in (0..7) : [A201635(n, k) for k in (0..n)]

A226013 Number of unimodal functions f:[n]->[2n] with f(1)<>1 and f(i)<>f(i+1).

Original entry on oeis.org

1, 1, 9, 70, 581, 4956, 43065, 379093, 3369301, 30168268, 271716644, 2459014504, 22342432139, 203682343840, 1862165051700, 17066961406095, 156758478514005, 1442549386731900, 13297258924349292, 122757267172891048, 1134800963513922996, 10503230892143398192

Offset: 0

Alois P. Heinz, May 22 2013

nonn

			a(0) = 1: [].
a(1) = 1: [2].
a(2) = 9: [2,1], [2,3], [2,4], [3,1], [3,2], [3,4], [4,1], [4,2], [4,3].
a(3) = 70: [2,3,1], [2,3,2], [2,3,4], ..., [6,5,2], [6,5,3], [6,5,4].
a(4) = 581: [2,3,2,1], [2,3,4,1], [2,3,4,2], ..., [8,7,6,3], [8,7,6,4], [8,7,6,5].

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A014301 (functions f:[n]->[n] with f(1)<>1 and f(i)<>f(i+1)).

a:= proc(n) option remember; `if`(n<3, (2*n-1)^2,
     ((49421666742*n^4 -205832874348*n^3 +295740702162*n^2
       -167673767628*n +29628103680) *a(n-1)
      +(27981954763*n^4 -127816385262*n^3 +231525900473*n^2
       -221063690262*n +102518080560) *a(n-2)
      +29529976*(2*n-5)*(4*n-9)*(n-3)*(4*n-11) *a(n-3))
      / (288*n*(2131486*n-3539195)*(3*n-4)*(3*n-2)))
    end:
seq(a(n), n=0..30);

a(n) ~ 2^(8*n-3/2) / (7*sqrt(Pi*n)*3^(3*n-3/2)). - Vaclav Kotesovec, Jul 16 2014

Recurrence (of order 2): 6*n*(3*n - 4)*(3*n - 2)*(77*n^2 - 244*n + 191)*a(n) = (37345*n^5 - 212742*n^4 + 463115*n^3 - 476646*n^2 + 228792*n - 40320)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(77*n^2 - 90*n + 24)*a(n-2). - Vaclav Kotesovec, Jul 16 2014

cp's OEIS Frontend

A201635 Triangle formed by T(n,n) = (-1)^n*Sum_{j=0..n} C(-n,j), T(n,k) = Sum_{j=0..k} T(n-1,j) for k=0..n-1, and n>=0, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Mathematica

PARI

Sage