cp's OEIS Frontend

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.

%I A283054 #30 Feb 29 2020 17:59:37
%S A283054 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,
%T A283054 197,1,8,34,101,232,428,625,626,1,9,43,144,376,804,1429,2055,2056,1,
%U A283054 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
%N 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.
%C A283054 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.
%C A283054 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.
%C A283054 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).
%C A283054 A closed-form formula for T(n,k) would be very desirable.
%H A283054 Ely Golden, <a href="/A283054/b283054.txt">Rows n = 0..124 of triangle, flattened</a>
%H A283054 Ely Golden, <a href="/A283054/a283054.java.txt">Java program for generating the triangle</a>
%H A283054 Ely Golden, <a href="/A283054/a283054_2.sagews.txt">Sage program for computing the polynomial of the n-th left diagonal</a>
%e A283054 First 7 rows:
%e A283054   1;
%e A283054   1,   2;
%e A283054   1,   3,   4;
%e A283054   1,   4,   8,   9;
%e A283054   1,   5,  13,  22,  23;
%e A283054   1,   6,  19,  41,  64,  65;
%e A283054   1,   7,  26,  67, 131, 196, 197;
%t A283054 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 *)
%o A283054 (SageMath)
%o A283054 def sideTriangleAt(a,b):
%o A283054     if(b==0): return 1
%o A283054     elif(b==a): return sideTriangleAt(a,b-1)+1
%o A283054     else: return sideTriangleAt(a,b-1)+sideTriangleAt(a-1,b)
%o A283054 def sideTriangle(size):
%o A283054     li=[]
%o A283054     for c in range(size):
%o A283054         for d in range(c+1):
%o A283054             if(d==0): li.append([1])
%o A283054             elif(d==c): li[c].append(li[c][d-1]+1)
%o A283054             else: li[c].append(li[c][d-1]+li[c-1][d])
%o A283054     return li
%o A283054 trig=sideTriangle(125)
%o A283054 for c in range(len(trig)):
%o A283054     print(str(trig[c])[1:-1].replace(",",""))
%o A283054 (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)
%o A283054 for(n=0,10,for(k=0,n,print1(T(n,k),", "))) \\ _Derek Orr_, Feb 28 2017
%K A283054 nonn,tabl
%O A283054 0,3
%A A283054 _Ely Golden_, Feb 27 2017