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 A014430 #57 Mar 03 2023 16:51:48
%S A014430 1,2,2,3,5,3,4,9,9,4,5,14,19,14,5,6,20,34,34,20,6,7,27,55,69,55,27,7,
%T A014430 8,35,83,125,125,83,35,8,9,44,119,209,251,209,119,44,9,10,54,164,329,
%U A014430 461,461,329,164,54,10,11,65,219,494,791,923,791,494,219,65,11
%N A014430 Subtract 1 from Pascal's triangle, read by rows.
%C A014430 Each value of the sequence (T(x,y)) is equal to the sum of all values in Pascal's Triangle that are in the rectangle defined by the tip (0,0) and the position (x,y). - Florian Kleedorfer (florian.kleedorfer(AT)austria.fm), May 23 2005
%C A014430 To clarify T(n,k) and A129696: We subtract I = Identity matrix from Pascal's triangle to obtain the beheaded variant, A074909. Then take column sums starting from the top of A074909 to get triangle A014430. Row sums of the inverse of triangle T(n,k) gives the Bernoulli numbers, A027641/A026642. Alternatively, triangle T(n,k) as an infinite lower triangular matrix * [the Bernoulli numbers as a vector] = [1, 1, 1, ...]. Given the B_n version starting (1, 1/2, 1/6, ...) triangle T(n,k) * the B_n vector [1, 1/2, 1/6, 0, -1/30, ...] = the triangular numbers. - _Gary W. Adamson_, Mar 13 2012
%C A014430 From _R. J. Mathar_, Apr 25 2016: (Start)
%C A014430 If regarded as a symmetric array of the form
%C A014430 1 2 3 4 5 ...
%C A014430 2 5 9 14 20 ...
%C A014430 3 9 19 34 55 ...
%C A014430 4 14 34 69 125 ...
%C A014430 5 20 55 125 251 ...
%C A014430 6 27 83 209 461 ...
%C A014430 7 35 119 329 791 ...
%C A014430 8 44 164 494 1286 ...
%C A014430 9 54 219 714 2001 ...
%C A014430 it contains the rows (and columns) A000096, A062748, A063258, A062988, A124089, ..., A035927 and so on and counts the multisets of digits of numbers in base b>=2 with d>=1 digits (equivalent to the comment in A035927). (End)
%C A014430 Proof of Florian Kleedorfer's formula: Take sums of the columns of the rectangle - these are all binomial coefficients by the Hockey Stick Identity. Note the locations of these coefficients: They form a row going almost all the way to the edge, only missing the 1 - apply the Hockey Stick Identity again. - _James East_, Jul 03 2020
%H A014430 Reinhard Zumkeller, <a href="/A014430/b014430.txt">Rows n=0..100 of triangle, flattened</a>
%F A014430 T(n, k) = T(n-1, k) + T(n-1, k-1) + 1, T(0, 0)=1. - _Ralf Stephan_, Jan 23 2005
%F A014430 G.f.: 1 / ((1-x)*(1-x*y)*(1-x*(1+y))). - _Ralf Stephan_, Jan 24 2005
%F A014430 T(n, k) = Sum_{j=0..k} Sum_{m=j..j+(n-k)} binomial(m, j). - Florian Kleedorfer (florian.kleedorfer(AT)austria.fm), May 23 2005
%F A014430 T(n, k) = binomial(n+2, k+1) - 1. - _G. C. Greubel_, Feb 25 2023
%e A014430 Triangle begins:
%e A014430 1;
%e A014430 2, 2;
%e A014430 3, 5, 3;
%e A014430 4, 9, 9, 4;
%e A014430 5, 14, 19, 14, 5;
%e A014430 6, 20, 34, 34, 20, 6;
%e A014430 7, 27, 55, 69, 55, 27, 7;
%e A014430 8, 35, 83, 125, 125, 83, 35, 8;
%t A014430 Table[Sum[Sum[Binomial[m, j], {m, j, j+(n-k)}], {j,0,k}], {n,0,10}, {k, 0,n}]//Flatten (* _Michael De Vlieger_, Sep 01 2020 *)
%t A014430 Table[Binomial[n+2,k+1] -1, {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 25 2023 *)
%o A014430 (Haskell)
%o A014430 a014430 n k = a014430_tabl !! n !! k
%o A014430 a014430_row n = a014430_tabl !! n
%o A014430 a014430_tabl = map (init . tail) $ drop 2 a014473_tabl
%o A014430 -- _Reinhard Zumkeller_, Apr 10 2012
%o A014430 (Magma) [Binomial(n+2,k+1)-1: k in [0..n], n in [0..13]]; // _G. C. Greubel_, Feb 25 2023
%o A014430 (SageMath) flatten([[binomial(n+2,k+1)-1 for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Feb 25 2023
%Y A014430 Cf. A000096, A007318, A014430, A026642, A027641, A027642, A035927.
%Y A014430 Cf. A062748, A063258, A062988, A074909, A124089, A124326, A129696.
%Y A014430 Triangle with zeros: A014473.
%Y A014430 Cf. A000295 (row sums).
%K A014430 nonn,tabl,easy
%O A014430 0,2
%A A014430 _N. J. A. Sloane_
%E A014430 More terms from _Erich Friedman_
%E A014430 Offset fixed by _Reinhard Zumkeller_, Apr 10 2012