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.
%I A228196 #76 Aug 24 2025 16:07:53
%S A228196 0,1,2,4,3,4,9,7,7,8,16,16,14,15,16,25,32,30,29,31,32,36,57,62,59,60,
%T A228196 63,64,49,93,119,121,119,123,127,128,64,142,212,240,240,242,250,255,
%U A228196 256,81,206,354,452,480,482,492,505,511,512,100,287,560,806,932,962,974,997,1016,1023,1024
%N A228196 A triangle formed like Pascal's triangle, but with n^2 on the left border and 2^n on the right border instead of 1.
%C A228196 The third row is (n^4 - n^2 + 24*n + 24)/12.
%C A228196 For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 04 2013
%H A228196 Boris Putievskiy, <a href="/A228196/b228196.txt">Rows n = 1..140 of triangle, flattened</a>
%H A228196 Rely Pellicer and David Alvo, <a href="http://www.academia.edu/956605/Modi_ed_Pascal_Triangle_and_Pascal_Surfaces">Modified Pascal Triangle and Pascal Surfaces</a> p.4
%H A228196 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A228196 T(n,0) = n^2, n>0; T(0,k) = 2^k; T(n, k) = T(n-1, k-1) + T(n-1, k) for n,k > 0. [corrected by _G. C. Greubel_, Nov 12 2019]
%F A228196 Closed-form formula for general case. Let L(m) and R(m) be the left border and the right border of Pascal like triangle, respectively. We denote binomial(n,k) by C(n,k).
%F A228196 As table read by antidiagonals T(n,k) = Sum_{m1=1..n} R(m1)*C(n+k-m1-1, n-m1) + Sum_{m2=1..k} L(m2)*C(n+k-m2-1, k-m2); n,k >=0.
%F A228196 As linear sequence a(n) = Sum_{m1=1..i} R(m1)*C(i+j-m1-1, i-m1) + Sum_{m2=1..j} L(m2)*C(i+j-m2-1, j-m2), where i=n-t*(t+1)/2-1, j=(t*t+3*t+4)/2-n-1, t=floor((-1+sqrt(8*n-7))/2); n>0.
%F A228196 Some special cases. If L(m)={b,b,b...} b*A000012, then the second sum takes form b*C(n+k-1,j). If L(m) is {0,b,2b,...} b*A001477, then the second sum takes form b*C(n+k,n-1). Similarly for R(m) and the first sum.
%F A228196 For this sequence L(m)=m^2 and R(m)=2^m.
%F A228196 As table read by antidiagonals T(n,k) = Sum_{m1=1..n} (2^m1)*C(n+k-m1-1, n-m1) + Sum_{m2=1..k} (m2^2)*C(n+k-m2-1, k-m2); n,k >=0.
%F A228196 As linear sequence a(n) = Sum_{m1=1..i} (2^m1)*C(i+j-m1-1, i-m1) + Sum_{m2=1..j} (m2^2)*C(i+j-m2-1, j-m2), where i=n-t*(t+1)/2-1, j=(t*t+3*t+4)/2-n-1, t=floor((-1+sqrt(8*n-7))/2).
%F A228196 As a triangular array read by rows, T(n,k) = Sum_{i=1..n-k} i^2*C(n-1-i, n-k-i) + Sum_{i=1..k} 2^i*C(n-1-i, k-i); n,k >=0. - _Greg Dresden_, Aug 06 2022
%e A228196 The start of the sequence as a triangular array read by rows:
%e A228196 0;
%e A228196 1, 2;
%e A228196 4, 3, 4;
%e A228196 9, 7, 7, 8;
%e A228196 16, 16, 14, 15, 16;
%e A228196 25, 32, 30, 29, 31, 32;
%e A228196 36, 57, 62, 59, 60, 63, 64;
%p A228196 T:= proc(n, k) option remember;
%p A228196 if k=0 then n^2
%p A228196 elif k=n then 2^k
%p A228196 else T(n-1, k-1) + T(n-1, k)
%p A228196 fi
%p A228196 end:
%p A228196 seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Nov 12 2019
%t A228196 T[n_, k_]:= T[n, k] = If[k==0, n^2, If[k==n, 2^k, T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 12 2019 *)
%t A228196 Flatten[Table[Sum[i^2 Binomial[n-1-i, n-k-i], {i,1,n-k}] + Sum[2^i Binomial[n-1-i, k-i], {i,1,k}], {n,0,10}, {k,0,n}]] (* _Greg Dresden_, Aug 06 2022 *)
%o A228196 (Python)
%o A228196 def funcL(n):
%o A228196 q = n**2
%o A228196 return q
%o A228196 def funcR(n):
%o A228196 q = 2**n
%o A228196 return q
%o A228196 for n in range (1,9871):
%o A228196 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A228196 i=n-t*(t+1)/2-1
%o A228196 j=(t*t+3*t+4)/2-n-1
%o A228196 sum1=0
%o A228196 sum2=0
%o A228196 for m1 in range (1,i+1):
%o A228196 sum1=sum1+funcR(m1)*binomial(i+j-m1-1,i-m1)
%o A228196 for m2 in range (1,j+1):
%o A228196 sum2=sum2+funcL(m2)*binomial(i+j-m2-1,j-m2)
%o A228196 sum=sum1+sum2
%o A228196 (PARI) T(n,k) = if(k==0, n^2, if(k==n, 2^k, T(n-1, k-1) + T(n-1, k) )); \\ _G. C. Greubel_, Nov 12 2019
%o A228196 (Sage)
%o A228196 @CachedFunction
%o A228196 def T(n, k):
%o A228196 if (k==0): return n^2
%o A228196 elif (k==n): return 2^n
%o A228196 else: return T(n-1, k-1) + T(n-1, k)
%o A228196 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 12 2019
%o A228196 (GAP)
%o A228196 T:= function(n,k)
%o A228196 if k=0 then return n^2;
%o A228196 elif k=n then return 2^n;
%o A228196 else return T(n-1,k-1) + T(n-1,k);
%o A228196 fi;
%o A228196 end;
%o A228196 Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 12 2019
%Y A228196 Cf. We denote Pascal-like triangle with L(n) on the left border and R(n) on the right border by (L(n),R(n)). A007318 (1,1), A008949 (1,2^n), A029600 (2,3), A029618 (3,2), A029635 (1,2), A029653 (2,1), A037027 (Fibonacci(n),1), A051601 (n,n) n>=0, A051597 (n,n) n>0, A051666 (n^2,n^2), A071919 (1,0), A074829 (Fibonacci(n), Fibonacci(n)), A074909 (1,n), A093560 (3,1), A093561 (4,1), A093562 (5,1), A093563 (6,1), A093564 (7,1), A093565 (8,1), A093644 (9,1), A093645 (10,1), A095660 (1,3), A095666 (1,4), A096940 (1,5), A096956 (1,6), A106516 (3^n,1), A108561(1,(-1)^n), A132200 (4,4), A134636 (2n+1,2n+1), A137688 (2^n,2^n), A160760 (3^(n-1),1), A164844(1,10^n), A164847 (100^n,1), A164855 (101*100^n,1), A164866 (101^n,1), A172171 (1,9), A172185 (9,11), A172283 (-9,11), A177954 (int(n/2),1), A193820 (1,2^n), A214292 (n,-n), A227074 (4^n,4^n), A227075 (3^n,3^n), A227076 (5^n,5^n), A227550 (n!,n!), A228053 ((-1)^n,(-1)^n), A228074 (Fibonacci(n), n).
%Y A228196 Cf. A000290 (row 1), A153056 (row 2), A000079 (column 1), A000225 (column 2), A132753 (column 3), A118885 (row sums of triangle array + 1), A228576 (generalized Pascal's triangle).
%K A228196 nonn,tabl,changed
%O A228196 1,3
%A A228196 _Boris Putievskiy_, Aug 15 2013
%E A228196 Cross-references corrected and extended by _Philippe Deléham_, Dec 27 2013