A283054 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1,k), T(n,0)=1, T(n,n) = T(n,n-1) + 1.

1, 1, 2, 1, 3, 4, 1, 4, 8, 9, 1, 5, 13, 22, 23, 1, 6, 19, 41, 64, 65, 1, 7, 26, 67, 131, 196, 197, 1, 8, 34, 101, 232, 428, 625, 626, 1, 9, 43, 144, 376, 804, 1429, 2055, 2056, 1, 10, 53, 197, 573, 1377, 2806, 4861, 6917, 6918, 1, 11, 64, 261, 834, 2211, 5017, 9878, 16795, 23713, 23714, 1, 12, 76, 337, 1171, 3382, 8399, 18277, 35072, 58785, 82499, 82500

Offset

0,3

Comments

The left diagonals form polynomial sequences. This is due to the observation that diagonal 0 D_0(x) = 1, and D_n(x) = D_n(x-1)+D_(n-1)(x+1), with D_n(-1) = 1 which is a recurrence that can be solved.

These polynomials begin 1, x+2, (x(x+7)+8)/2, (x(x(x+15)+62)+54)/6, (x(x(x(x+26)+227)+730)+552)/24, etc., the first 3 of which correspond to A000012(n), A000027(n+2), and A034856(n+2), respectively.

The rightmost diagonal appears to follow A014137(n). The second rightmost appears to follow A014138(n+1), the third appears to follow A001453(n+2), the fourth appears to follow A114277(n), and the fifth appears to follow A143955(n+3).

A closed-form formula for T(n,k) would be very desirable.

Links

Ely Golden, Rows n = 0..124 of triangle, flattened

Ely Golden, Java program for generating the triangle

Ely Golden, Sage program for computing the polynomial of the n-th left diagonal

Example

First 7 rows:
  1;
  1,   2;
  1,   3,   4;
  1,   4,   8,   9;
  1,   5,  13,  22,  23;
  1,   6,  19,  41,  64,  65;
  1,   7,  26,  67, 131, 196, 197;

Mathematica

T[0, 0] = 1; T[n_, k_] := T[n, k] = Which[k == 0, 1, k == n, T[n, n - 1] + 1, True, T[n, k - 1] + T[n - 1, k]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 27 2017 *)

Prog

(SageMath)
def sideTriangleAt(a, b):
    if(b==0): return 1
    elif(b==a): return sideTriangleAt(a, b-1)+1
    else: return sideTriangleAt(a, b-1)+sideTriangleAt(a-1, b)
def sideTriangle(size):
    li=[]
    for c in range(size):
        for d in range(c+1):
            if(d==0): li.append([1])
            elif(d==c): li[c].append(li[c][d-1]+1)
            else: li[c].append(li[c][d-1]+li[c-1][d])
    return li
trig=sideTriangle(125)
for c in range(len(trig)):
    print(str(trig[c])[1:-1].replace(", ", ""))
(PARI) T(n, k)=if(k==0, return(1)); if(k==n, return(T(n, n-1)+1)); T(n, k-1)+T(n-1, k)
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ Derek Orr, Feb 28 2017

Crossrefs

Sequence in context: A208341 A201634 A210211 * A247358 A297224 A180383

Adjacent sequences: A283051 A283052 A283053 * A283055 A283056 A283057

Keyword

nonn,tabl

Author

Ely Golden, Feb 27 2017

Status

approved