A104712 - OEIS

%I #76 Feb 13 2022 23:18:02

%S 1,3,1,6,4,1,10,10,5,1,15,20,15,6,1,21,35,35,21,7,1,28,56,70,56,28,8,

%T 1,36,84,126,126,84,36,9,1,45,120,210,252,210,120,45,10,1,55,165,330,

%U 462,462,330,165,55,11,1,66,220,495,792,924,792,495,220,66,12,1,78,286,715

%N Pascal's triangle, with the first two columns removed.

%C A000295 (Eulerian numbers) gives the row sums.

%C Write A004736 and Pascal's triangle as infinite lower triangular matrices A and B; then A*B is this triangle.

%C From _Peter Luschny_, Apr 10 2011: (Start)

%C A slight variation has a combinatorial interpretation: remove the last column and the second one from Pascal's triangle. Let P(m, k) denote the set partitions of {1,2,..,n} with the following properties:

%C (a) Each partition has at least one singleton block;

%C (c) k is the size of the largest block of the partition;

%C (b) m = n - k + 1 is the number of parts of the partition.

%C Then A000295(n) = Sum_{k=1..n} card(P(n-k+1,k)).

%C For instance, A000295(4) = P(4,1) + P(3,2) + P(2,3) + P(1,4) = card({1|2|3|4}) + card({1|2|34, 1|3|24,1|4|23, 2|3|14, 2|4|13, 3|4|12}) + card({1|234, 2|134, 3|124, 4|123}) = 1 + 6 + 4 = 11.

%C This interpretation can be superimposed on the sequence by changing the offset to 1 and adding the value 1 in front. The triangle then starts

%C   1;

%C   1,  3;

%C   1,  6,  4;

%C   1, 10, 10,  5;

%C   1, 15, 20, 15,  6;

%C   ...

%C (End)

%C Diagonal sums are A001924(n+1). - _Philippe Deléham_, Jan 11 2014

%C Relation to K-theory: T acting on the column vector (d,-d^2,d^3,...) generates the Euler classes for a hypersurface of degree d in CP^n. Cf. Dugger p. 168, A111492, A238363, and A135278. - _Tom Copeland_, Apr 11 2014

%H G. C. Greubel, <a href="/A104712/b104712.txt">Rows n=2..100 of triangle, flattened</a>

%H D. Dugger, <a href="http://math.uoregon.edu/~ddugger/kgeom.pdf">A Geometric Introduction to K-Theory</a>

%H Candice A. Marshall, <a href="https://mdsoar.org/handle/11603/10353">Construction of Pseudo-Involutions in the Riordan Group</a>, Dissertation, Morgan State University, 2017.

%H T. Saito, <a href="http://arxiv.org/abs/1110.1717">The discriminant and the determinant of a hypersurface of even dimension</a> (p. 4), arXiv:1110.1717 [math.AG], 2011-2012.

%F a(n,k) = binomial(n,k), for 2 <= k <= n.

%F From _Peter Bala_, Jul 16 2013: (Start)

%F The following remarks assume an offset of 0.

%F Riordan array (1/(1 - x)^3, x/(1 - x)).

%F O.g.f.: 1/(1 - t)^2*1/(1 - (1 + x)*t) = 1 + (3 + x)*t + (6 + 4*x + x^2)*t^2 + ....

%F E.g.f.: (1/x*d/dt)^2 (exp(t)*(exp(x*t) - 1 - x*t) = 1 + (3 + x)*t + (6 + 4*x + x^2)*t^2/2! + ....

%F The infinitesimal generator for this triangle has the sequence [3,4,5,...] on the main subdiagonal and 0's elsewhere. (End)

%F As triangle T(n,k), 0<=k<=n: T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham_, Jan 11 2014

%F From _Tom Copeland_, Apr 11 2014: (Start)

%F A) The infinitesimal generator for this matrix is given in A132681 with m=2. See that entry for numerous relations to differential operators and the Laguerre polynomials of order m=2, i.e., Lag(n,t,2) = Sum_{j=0..n} binomial(n+2,n-j)*(-t)^j/j!.

%F B) O.g.f.: 1 / { [ 1 - t * x/(1-x) ] * (1-x)^3 }

%F C) O.g.f. of row e.g.f.s: exp[t*x/(1-x)]/(1-x)^3 = [Sum_{n>=0} x^n * Lag(n,-t,2)] = 1 + (3 + t)*x + (6 + 4t + t^2/2!)*x^2 + (10 + 10t + 5t^2/2! + t^3/3!)*x^3 + ....

%F D) E.g.f. of row o.g.f.s: [(1+t)*exp((1+t)*x) - (1+t+t*x)exp(x)]/t^2. (End)

%F O.g.f. for m-th row (m=n-2): [(1+x)^(m+2)-(1+(m+2)*x)]/x^2. - _Tom Copeland_, Apr 16 2014

%F Reverse T = [St2]*dP*[St1]- dP = [St2]*(exp(x*M)-I)*[St1]-(exp(x*M)-I) with top two rows of zeros removed, [St1]=padded A008275 just as [St2]=A048993=padded A008277, dP= A132440, M=A238385-I, and I=identity matrix. Cf. A238363. - _Tom Copeland_, Apr 26 2014

%F O.g.f. of column k (with k leading zeros): (x^k)/(1-x)^(k+1), k >= 2. - _Wolfdieter Lang_, Mar 20 2015

%e The triangle a(n, k) begins:

%e n\k  2   3   4    5    6    7    8   9  10 11 12 13

%e 2:   1

%e 3:   3   1

%e 4:   6   4   1

%e 5:  10  10   5    1

%e 6:  15  20  15    6    1

%e 7:  21  35  35   21    7    1

%e 8:  28  56  70   56   28    8    1

%e 9:  36  84 126  126   84   36    9   1

%e 10: 45 120 210  252  210  120   45  10   1

%e 11: 55 165 330  462  462  330  165  55  11  1

%e 12: 66 220 495  792  924  792  495 220  66 12  1

%e 13: 78 286 715 1287 1716 1716 1287 715 286 78 13  1

%e ... reformatted. - _Wolfdieter Lang_, Mar 20 2015

%t t[n_, k_] := Binomial[n, k]; Table[ t[n, k], {n, 2, 13}, {k, 2, n}] // Flatten (* _Robert G. Wilson v_, Apr 16 2011 *)

%o (PARI) for(n=2, 10, for(k=2,n, print1(binomial(n,k), ", "))) \\ _G. C. Greubel_, May 15 2018

%o (Magma) /* As triangle */ [[Binomial(n, k): k in [2..n]]: n in [2..10]]; // _G. C. Greubel_, May 15 2018

%Y Cf. A000295, A007318, A008292, A104713, A027641/A027642 (first Bernoulli numbers B-), A164555/A027642 (second Bernoulli numbers B+), A176327/A176289.

%K nonn,tabl,easy

%O 2,2

%A _Gary W. Adamson_, Mar 19 2005

%E Edited and extended by _David Wasserman_, Jul 03 2007