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A104712 Pascal's triangle, with the first two columns removed. 17
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, 1, 36, 84, 126, 126, 84, 36, 9, 1, 45, 120, 210, 252, 210, 120, 45, 10, 1, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1, 78, 286, 715 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

A000295 (Eulerian numbers) gives the row sums.

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

From Peter Luschny, Apr 10 2011: (Start)

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:

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

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

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

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

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.

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

  1

  1,  3

  1,  6,  4

  1, 10, 10,  5

  1, 15, 20, 15,  6

(End)

Diagonal sums are A001924(n+1). - Philippe Deléham, Jan 11 2014

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

LINKS

Table of n, a(n) for n=2..70.

D. Dugger, A Geometric Introduction to K-Theory

T. Saito, The discriminant and the determinant of a hypersurface of even dimension (p. 4), arXiv:1110.1717 [math.AG], 2011-2012.

FORMULA

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

From Peter Bala, Jul 16 2013: (Start)

The following remarks assume an offset of 0.

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

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

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! + ....

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

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

From Tom Copeland, Apr 11 2014: (Start)

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!.

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

C) O.g.f of row e.g.f.s: exp[t*x/(1-x)]/(1-x)^3 = [sum(n=0,1,...) 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 + ....

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)

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

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

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

EXAMPLE

The triangle a(n, k) begins:

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

2:   1

3:   3   1

4:   6   4   1

5:  10  10   5    1

6:  15  20  15    6    1

7:  21  35  35   21    7    1

8:  28  56  70   56   28    8    1

9:  36  84 126  126   84   36    9   1

10: 45 120 210  252  210  120   45  10   1

11: 55 165 330  462  462  330  165  55  11  1

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

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

... reformatted. - Wolfdieter Lang, Mar 20 2015

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A286158 A185915 A086270 * A122177 A255874 A108286

Adjacent sequences:  A104709 A104710 A104711 * A104713 A104714 A104715

KEYWORD

nonn,tabl,easy

AUTHOR

Gary W. Adamson, Mar 19 2005

EXTENSIONS

Edited and extended by David Wasserman, Jul 03 2007

STATUS

approved

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Last modified June 28 04:43 EDT 2017. Contains 288813 sequences.