login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A228576 A triangle formed like generalized Pascal's triangle. The rule is T(n,k) = 2*T(n-1,k-1) + T(n-1,k), the left border is n and the right border is n^2 instead of 1. 19

%I

%S 0,1,1,2,3,4,3,7,10,9,4,13,24,29,16,5,21,50,77,74,25,6,31,92,177,228,

%T 173,36,7,43,154,361,582,629,382,49,8,57,240,669,1304,1793,1640,813,

%U 64,9,73,354,1149,2642,4401,5226,4093,1690,81,10,91,500,1857,4940,9685,14028,14545,9876,3461,100

%N A triangle formed like generalized Pascal's triangle. The rule is T(n,k) = 2*T(n-1,k-1) + T(n-1,k), the left border is n and the right border is n^2 instead of 1.

%H Boris Putievskiy, <a href="/A228576/b228576.txt">Rows n = 1..140 of triangle, flattened</a>

%H Rely Pellicer, David Alvo, <a href="http://www.academia.edu/956605/Modified_Pascal_Triangle_and_Pascal_Surfaces">Modified Pascal Triangle and Pascal Surfaces</a> p.4

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.

%H Rattanapol Wasutharat, Kantaphon Kuhapatanakul, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/41-44-2012/wasutharatIJCMS41-44-2012.pdf">The Generalized Pascal-Like Triangle and Applications</a> Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 41, pp. 1989 - 1992

%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F T(n, k) = 2*T(n-1, k-1) + T(n-1, k) for n,k >=0, with T(n,0) = n, T(n,n) = n^2.

%F Closed-form formula for generalized Pascal's triangle. Let a,b be any numbers. The rule is T(n, k) = a*T(n-1, k-1) + b*T(n-1, k) for n,k >0. Let L(m) and R(m) be the left border and the right border generalized Pascal's triangle, respectively.

%F As table read by antidiagonals T(n,k) = Sum_{m1=1..n} a^(n-m1) * b^k*R(m1)*C(n+k-m1-1,n-m1) + Sum_{m2=1..k} a^n*b^(k-m2)*L(m2)*C(n+k-m2-1,k-m2); n,k >=0.

%F As linear sequence a(n) = Sum_{m1=1..i} a^(i-m1)*b^j*R(m1)*C(i+j-m1-1,i-m1) + Sum_{m2=1..j} a^i*b^(j-m2)*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 Some special cases. If a=b=1, then the closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196.

%F If a=0, then as table read by antidiagonals T(n,k)=b*R(n), as linear sequence a(n)=b*R(i), where i=n-t*(t+1)/2-1, t=floor((-1+sqrt(8*n-7))/2); n>0. The sequence a(n) is the reluctant sequence of sequence b*R(n) - a(n) is triangle array read by rows: row number k coincides with first k elements of the sequence b*R(n). Similarly for b=0, we get T(n,k)=a*L(k).

%F For this sequence L(m)=m and R(m)=m^2, a=2, b=1. As table read by antidiagonals T(n,k) = Sum_{m1=1..n} 2^(n-m1)*m1**2*C(n+k-m1-1,n-m1) + Sum_{m2=1..k} 2^n*m2*C(n+k-m2-1,k-m2); n,k >=0.

%F As linear sequence a(n) = Sum_{m1=1..i} 2^(i-m1)*m1**2*C(i+j-m1-1, i-m1) + Sum_{m2=1..j} 2^i*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.

%e The start of the sequence as triangle array read by rows:

%e 0;

%e 1, 1;

%e 2, 3, 4;

%e 3, 7, 10, 9;

%e 4, 13, 24, 29, 16;

%e 5, 21, 50, 77, 74, 25;

%e ...

%p T := proc(n, k) option remember;

%p if k = 0 then RETURN(n) fi;

%p if k = n then RETURN(n^2) fi;

%p 2*T(n-1, k-1) + T(n-1, k) end:

%p seq(seq(T(n,k),k=0..n),n=0..9); # _Peter Luschny_, Aug 26 2013

%t T[n_, 0]:= n; T[n_, n_]:= n^2; T[n_, k_]:= T[n, k] = 2*T[n-1, k-1]+T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-Fran├žois Alcover_, Feb 25 2014 *)

%o (PARI) T(n,k) = if(k==0, n, if(k==n, n^2, 2*T(n-1, k-1) + T(n-1, k) )); \\ _G. C. Greubel_, Nov 13 2019

%o (MAGMA)

%o function T(n,k)

%o if k eq 0 then return n;

%o elif k eq n then return n^2;

%o else return 2*T(n-1,k-1) + T(n-1,k);

%o end if;

%o return T;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 13 2019

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (k==0): return n

%o elif (k==n): return n^2

%o else: return 2*T(n-1,k-1) + T(n-1, k)

%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 13 2019

%o (GAP)

%o T:= function(n,k)

%o if k=0 then return n;

%o elif k=n then return n^2;

%o else return 2*T(n-1,k-1) + T(n-1,k);

%o fi;

%o end;

%o Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 13 2019

%Y Cf. We denote generalized Pascal's like triangle with coefficients a, b and with L(n) on the left border and R(n) on the right border by (a,b,L(n),R(n)). The list of sequences for (1,1,L(n),R(n)) see A228196;

%Y A038207 (1,2,2^n,1), A105728 (1, 2, 1, n+1), A112468 (1,-1,1,1), A112626 (1,2,3^n,1), A119258 (2,1,1,1), A119673 (3,1,1,1), A119725 (3,2,1,1), A119726 (4,2,1,1), A119727 (5,2,1,1), A209705 (2,1,n+1,0);

%Y A002061 (column 2), A000244 (sums of rows r of triangle array - (r-2)(r+1)/2).

%K nonn,tabl

%O 1,4

%A _Boris Putievskiy_, Aug 26 2013

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 05:14 EST 2019. Contains 329839 sequences. (Running on oeis4.)