A174294 Triangle T(n,k), read by rows, T(n,k) = (T(n-1,k-1) + T(n-2,k-1)) - (T(n-1,k) + T(n-2,k)), with T(n, 0) = T(n, k) = 1 and T(n, 1) = (n mod 2).

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, -1, 0, 1, 1, 0, 0, 2, -2, 0, 1, 1, 1, 0, 0, 3, -3, 0, 1, 1, 0, 1, -2, 1, 4, -4, 0, 1, 1, 1, 0, 3, -6, 3, 5, -5, 0, 1, 1, 0, 0, 0, 6, -12, 6, 6, -6, 0, 1, 1, 1, 1, -3, 3, 9, -20, 10, 7, -7, 0, 1, 1, 0, 0, 4, -12, 12, 11, -30, 15, 8, -8, 0, 1

Offset

0,25

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n,k) = (T(n-1,k-1) + T(n-2,k-1)) - (T(n-1,k) + T(n-2,k)), with T(n, 0) = T(n, k) = 1 and T(n, 1) = (n mod 2).

Example

Table begins:
  n\k|...0...1...2...3...4...5...6...7...8...9..10
  ---|--------------------------------------------
  0..|...1
  1..|...1...1
  2..|...1...0...1
  3..|...1...1...0...1
  4..|...1...0...0...0...1
  5..|...1...1...1..-1...0...1
  6..|...1...0...0...2..-2...0...1
  7..|...1...1...0...0...3..-3...0...1
  8..|...1...0...1..-2...1...4..-4...0...1
  9..|...1...1...0...3..-6...3...5..-5...0...1
  10.|...1...0...0...0...6.-12...6...6..-6...0...1

Mathematica

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, If[k==1, Mod[n, 2], T[n-1, k-1] +T[n-2, k-1] -T[n-1, k] -T[n-2, k] ]]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 25 2021 *)

Prog

(Sage)
@CachedFunction
def T(n, k): # A174294
    if (k<0 or k>n): return 0
    elif (k==0 or k==n): return 1
    elif (k==1): return n%2
    else: return T(n-1, k-1) + T(n-2, k-1) - T(n-1, k) - T(n-2, k)
flatten([[T(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 25 2021

Crossrefs

Cf. A112467, A112468, A174294, A174295, A174296, A174297.

Keyword

sign,tabl

Author

Mats Granvik, Mar 15 2010

Status

approved

A114700 Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I), where T(n,k) = [T^-1](n-1,k) + [T^-1](n-1,k-1) for n>k>0, with T(n,0)=T(n,n)=1 for n>=0 and I is the identity matrix.

1, 1, 1, 1, 0, 1, 1, -1, 1, 1, 1, 0, 0, 0, 1, 1, -1, 0, 0, 1, 1, 1, 0, 1, 0, -1, 0, 1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 0, 2, 2, 0, -2, -2, 0, 1, 1, -1, -2, -4, -2, 2, 4, 2, 1, 1, 1, 0, 3, 6, 6, 0, -6, -6, -3, 0, 1, 1, -1, -3, -9, -12, -6, 6, 12, 9, 3, 1, 1, 1, 0, 4, 12, 21, 18, 0, -18, -21, -12, -4, 0, 1

Offset

0,39

Comments

The rows of this triangle are symmetric up to sign. Row sums = 2 after row 0. Unsigned row sums = A116466. Row squared sums = A116467. Central terms of odd rows: T(2*n+1,n+1) = |A064310(n)|.

Links

Table of n, a(n) for n=0..90.

Formula

G.f.: A(x,y) = 1/(1-x*y)+ x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y)/(1-x*y). G.f. of matrix power T^m: 1/(1-x*y)+ m*x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y)/(1-x*y).

Example

Matrix inverse is: T^-1 = 2*I - T.
Matrix log is: log(T) = T - I.
Triangle T begins:
1;
1, 1;
1, 0, 1;
1,-1, 1, 1;
1, 0, 0, 0, 1;
1,-1, 0, 0, 1, 1;
1, 0, 1, 0,-1, 0, 1;
1,-1,-1,-1, 1, 1, 1, 1;
1, 0, 2, 2, 0,-2,-2, 0, 1;
1,-1,-2,-4,-2, 2, 4, 2, 1, 1;
1, 0, 3, 6, 6, 0,-6,-6,-3, 0, 1;
1,-1,-3,-9,-12,-6, 6, 12, 9, 3, 1, 1;
1, 0, 4, 12, 21, 18, 0,-18,-21,-12,-4, 0, 1; ...
The g.f. of column k, C_k(x), obeys the recurrence:
C_k = C_{k-1} + (-1)^k*x*(1+2*x)/(1-x)/(1+x)^k with C_0 = 1/(1-x);
so that column g.f.s continue as:
C_1 = C_0 - x*(1+2*x)/(1-x)/(1+x),
C_2 = C_1 + x*(1+2*x)/(1-x)/(1+x)^2,
C_3 = C_2 - x*(1+2*x)/(1-x)/(1+x)^3, ...

Prog

(PARI) T(n, k)=local(x=X+X*O(X^n), y=Y+Y*O(Y^k)); polcoeff(polcoeff( 1/(1-x*y)+ x*(1+x-2*x^2*y)/(1-x)/(1+x+x*y)/(1-x*y), n, X), k, Y)
(PARI) T(n, k)=local(M=matrix(n+1, n+1)); for(r=1, n+1, for(c=1, r, M[r, c]=if(r==c, 1, if(c==1, 1, if(c>1, (2*M^0-M)[r-1, c-1])+(2*M^0-M)[r-1, c])))); return(M[n+1, k+1])

Crossrefs

Cf. A116466 (unsigned row sums), A116467 (row squared sums), A064310 (central terms); A112555 (variant).

Keyword

sign,tabl

Author

Paul D. Hanna, Feb 19 2006

Status

approved

A276321 List of B-spline interpolation matrices M(n,i,j) of orders n >= 1 read by rows (0 <= i < n, 0 <= j < n).

1, -1, 1, 1, 0, 1, -2, 1, -2, 2, 0, 1, 1, 0, -1, 3, -3, 1, 3, -6, 3, 0, -3, 0, 3, 0, 1, 4, 1, 0, 1, -4, 6, -4, 1, -4, 12, -12, 4, 0, 6, -6, -6, 6, 0, -4, -12, 12, 4, 0, 1, 11, 11, 1, 0, -1, 5, -10, 10, -5, 1, 5, -20, 30, -20, 5, 0, -10, 20, 0, -20, 10, 0, 10

Offset

1,7

Comments

B-spline interpolation matrices of orders n >= 1. Each matrix M(i, j) is of dimensions n X n.

Links

T .C. I. Niessink, Matrices for n = 1..31

J. I. Craig, Notes on B-spline development, (spring 2004), 7.

T. C. I. Niessink, C source code, bigint (n >= 1).

T. C. I. Niessink, C source code, int (1 <= n <= 12).

T. C. I. Niessink, C source code, long long (1 <= n <= 20).

Wikipedia, B-spline.

Formula

M(n, i, j) = C(n-1, i) * Sum_{m=j..n-1} (n-(m+1))^i * (-1)^(m-j) * C(n, m-j), where binomial coefficient C(n, k) = n! / (k!*(n-k)!).

Example

The B-spline interpolation matrices for orders n = 1..4 are:
--
1
--
-1  1
1  0
--
1 -2  1
-2  2  0
1  1  0
--
-1  3 -3  1
3 -6  3  0
-3  0  3  0
1  4  1  0
--
These values can be used when implementing interpolation using b-splines. This is what the algorithm for order n = 2 (linear interpolation) looks like in pseudocode:
--
a0 = -1 * p[i] + 1 * p[i+1]
a1 =  1 * p[i] + 0 * p[i+1]
y = (u*a0 + a1) / (n-1)!
--
where u is in the [0, 1) range. And here is another example for order n = 4 (cubic interpolation):
--
a0 = -1 * p[i] +  3 * p[i+1] + -3 * p[i+2] + 1 * p[i+3]
a1 =  3 * p[i] + -6 * p[i+1] +  3 * p[i+2] + 0 * p[i+3]
a2 = -3 * p[i] +  0 * p[i+1] +  3 * p[i+2] + 0 * p[i+3]
a3 =  1 * p[i] +  4 * p[i+1] +  1 * p[i+2] + 0 * p[i+3]
y = (u^3*a0 + u^2*a1 + u*a2 + a3) / (n-1)!
--
You can optimize the algorithm by dividing all numbers in the matrix by (n-1)!, and then rewriting y as:
--
y = u*(u*(u*a0 + a1) + a2) + a3

Prog

(C)
int fact(int n)
{
  int y = 1;
  int i;
  for (i = 1; i <= n; ++i) y *= i;
  return y;
}
int ipow(int n, int p)
{
  int y = 1;
  int i;
  for (i = 0; i < p; ++i) y *= n;
  return y;
}
int C(int n, int k)
{
  return fact(n) / (fact(k) * fact(n - k));
}
int M(int n, int i, int j)
{
  int y = 0;
  int m;
  for (m = j; m < n; ++m)
  {
    y += ipow(n - (m + 1), i) * ipow(-1, m - j) * C(n, m - j);
  }
  return C(n - 1, i) * y;
}
(PARI) M(n, i, j) = binomial(n-1, i) * sum(m=j, n-1, (n-(m+1))^i * (-1)^(m-j) * binomial(n, m-j));
doM(n) = for (i=0, n-1, for (j=0, n-1, print1(M(n, i, j), ", ")); print());
tabf(nn) = for (n=1, nn, doM(n)); \\ Michel Marcus, Sep 06 2016

Crossrefs

Cf. A007318.

Keyword

sign,tabf

Author

Theo Niessink, Aug 30 2016

Status

approved

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} j*x^(j*i))^2.

1, 1, 0, 1, -2, 0, 1, -2, 1, 0, 1, -2, -3, -2, 0, 1, -2, -3, 10, 4, 0, 1, -2, -3, 4, -4, -4, 0, 1, -2, -3, 4, 14, -20, 5, 0, 1, -2, -3, 4, 6, -8, 41, -6, 0, 1, -2, -3, 4, 6, 16, -46, 2, 9, 0, 1, -2, -3, 4, 6, 6, -30, 14, -111, -12, 0, 1, -2, -3, 4, 6, 6, 0, -58

Offset

0,5

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Example

Square array begins:
   1,  1,   1,  1,  1, ...
   0, -2,  -2, -2, -2, ...
   0,  1,  -3, -3, -3, ...
   0, -2,  10,  4,  4, ...
   0,  4,  -4, 14,  6, ...
   0, -4, -20, -8, 16, ...

Crossrefs

Columns k=0..1 give A000007, A022597.

Rows n=0 gives A000012.

Main diagonal gives A252650.

Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^m: this sequence (m=-2), A290217 (m=-1), A290216 (m=1), A293377 (m=2).

Keyword

sign,tabl

Author

Seiichi Manyama, Oct 07 2017

Status

approved

A054054 Smallest digit of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 0, 1, 2, 3, 4, 5, 6, 6, 6, 6, 0, 1, 2, 3, 4, 5, 6, 7, 7, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 0, 0, 0, 0

Offset

0,3

Comments

a(n) = 0 for almost all n. - Charles R Greathouse IV, Oct 02 2013

More precisely, a(n) = 0 asymptotically almost surely, i.e., except for a set of density 0: As the number of digits of n grows, the probability of having no zero digit goes to zero as 0.9^(length of n), although there are infinitely many counterexamples. - M. F. Hasler, Oct 11 2015

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Formula

a(A011540(n)) = 0; a(A052382(n)) > 0. - Reinhard Zumkeller, Apr 25 2012

a(n) = A262188(n,0). - Reinhard Zumkeller, Sep 14 2015

a(n) = 0 iff A007954(n) = 0. - M. F. Hasler, Oct 11 2015

a(n) = 9 - A054055(A061601(n)). - Robert Israel, Jul 07 2016

Example

a(12) = 1 because 1 < 2.

Maple

seq(min(convert(n, base, 10)), n=0..100); # Robert Israel, Jul 07 2016

Mathematica

A054054[n_]:=Min[IntegerDigits[n]]

Prog

(Haskell)
a054054 = f 9 where
   f m x | x <= 9 = min m x
         | otherwise = f (min m d) x' where (x', d) = divMod x 10
-- Reinhard Zumkeller, Jun 20 2012, Apr 25 2012
(PARI) A054054(n)=if(n, vecmin(digits(n)))  \\ or: Set(digits(n))[1]. - M. F. Hasler, Jan 23 2013

Crossrefs

Cf. A054055.

Cf. A061601, A011540, A052382, A262188.

Keyword

base,easy,nonn

Author

Henry Bottomley, Apr 29 2000

Extensions

Edited by M. F. Hasler, Oct 11 2015

Status

approved

A072574 Triangle T(n,k) of number of compositions (ordered partitions) of n into exactly k distinct parts, 1<=k<=n.

1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 1, 4, 0, 0, 0, 1, 4, 6, 0, 0, 0, 1, 6, 6, 0, 0, 0, 0, 1, 6, 12, 0, 0, 0, 0, 0, 1, 8, 18, 0, 0, 0, 0, 0, 0, 1, 8, 24, 24, 0, 0, 0, 0, 0, 0, 1, 10, 30, 24, 0, 0, 0, 0, 0, 0, 0, 1, 10, 42, 48, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 48, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 60, 120, 0

Offset

1,5

Comments

If terms in the compositions did not need to be distinct then the triangle would have values C(n-1,k-1), essentially A007318 offset.

Links

Joerg Arndt, Table of n, a(n) for n = 1..5050 (rows 1..100, flattened).

B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97.

Index entries for sequences related to compositions

Formula

T(n, k) = T(n-k, k)+k*T(n-k, k-1) [with T(n, 0)=1 if n=0 and 0 otherwise] = A000142(k)*A060016(n, k).

G.f.: sum(n>=0, n! * z^n * q^((n^2+n)/2) / prod(k=1..n, 1-q^k ) ), rows by powers of q, columns by powers of z; includes row 0 (drop term for n=0 for this triangle, see PARI code); setting z=1 gives g.f. for A032020. [Joerg Arndt, Oct 20 2012]

Example

T(6,2)=4 since 6 can be written as 1+5=2+4=4+2=5+1.
Triangle starts (trailing zeros omitted for n>=10):
[ 1]  1;
[ 2]  1, 0;
[ 3]  1, 2, 0;
[ 4]  1, 2, 0, 0;
[ 5]  1, 4, 0, 0, 0;
[ 6]  1, 4, 6, 0, 0, 0;
[ 7]  1, 6, 6, 0, 0, 0, 0;
[ 8]  1, 6, 12, 0, 0, 0, 0, 0;
[ 9]  1, 8, 18, 0, 0, 0, 0, 0, 0;
[10]  1, 8, 24, 24, 0, 0, ...;
[11]  1, 10, 30, 24, 0, 0, ...;
[12]  1, 10, 42, 48, 0, 0, ...;
[13]  1, 12, 48, 72, 0, 0, ...;
[14]  1, 12, 60, 120, 0, 0, ...;
[15]  1, 14, 72, 144, 120, 0, 0, ...;
[16]  1, 14, 84, 216, 120, 0, 0, ...;
[17]  1, 16, 96, 264, 240, 0, 0, ...;
[18]  1, 16, 114, 360, 360, 0, 0, ...;
[19]  1, 18, 126, 432, 600, 0, 0, ...;
[20]  1, 18, 144, 552, 840, 0, 0, ...;
These rows (without the zeros) are shown in the Richmond/Knopfmacher reference.
From Gus Wiseman, Oct 17 2022: (Start)
Column n = 8 counts the following compositions.
  (8)  (1,7)  (1,2,5)
       (2,6)  (1,3,4)
       (3,5)  (1,4,3)
       (5,3)  (1,5,2)
       (6,2)  (2,1,5)
       (7,1)  (2,5,1)
              (3,1,4)
              (3,4,1)
              (4,1,3)
              (4,3,1)
              (5,1,2)
              (5,2,1)
(End)

Mathematica

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], UnsameQ@@#&], Length[#]==k&]], {n, 0, 15}, {k, 1, n}] (* Gus Wiseman, Oct 17 2022 *)

Prog

(PARI)
N=21;  q='q+O('q^N);
gf=sum(n=0, N, n! * z^n * q^((n^2+n)/2) / prod(k=1, n, 1-q^k ) );
/* print triangle: */
gf -= 1; /* remove row zero */
P=Pol(gf, 'q);
{ for (n=1, N-1,
    p = Pol(polcoeff(P, n), 'z);
    p += 'z^(n+1);  /* preserve trailing zeros */
    v = Vec(polrecip(p));
    v = vector(n, k, v[k]); /* trim to size n */
    print(v);
); }
/* Joerg Arndt, Oct 20 2012 */

Crossrefs

Columns (offset) include A057427 and A052928.

Row sums are A032020.

Cf. A060016, A072576.

A008289 is the version for partitions (zeros removed).

A072575 counts strict compositions by maximum.

A097805 is the non-strict version, or A007318 (zeros removed).

A113704 is the constant instead of strict version.

A216652 is a condensed version (zeros removed).

A336131 counts splittings of partitions with distinct sums.

A336139 counts strict compositions of each part of a strict composition.

Cf. A075900, A097910, A307068, A336127, A336128, A336130, A336132, A336141, A336142, A336342, A336343.

Keyword

nonn,tabl

Author

Henry Bottomley, Jun 21 2002

Status

approved

A056557 Second tetrahedral coordinate.

0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5

Offset

0,8

Comments

If {(X,Y,Z)} are triples of nonnegative integers with X >= Y >= Z ordered by X, Y and Z, then X=A056556(n), Y=A056557(n) and Z=A056558(n)

Links

Table of n, a(n) for n=0..104.

Formula

a(n) = floor((sqrt(8*(n-A056556(n)*(A056556(n)+1)*(A056556(n)+2)/6)+1)-1)/2) = A003056(n-A000292(A056556(n)-1)).

a(n+1) = 0 if A056556(n) = A056558(n); a(n+1) = a(n)+1 if a(n) = A056558(n); otherwise a(n+1) = a(n). - Graeme McRae, Jan 09 2007

Crossrefs

Cf. A000292, A002262, A003056, A056556, A056558, A056559, A056560.

See also A194847, A194848.

Keyword

nonn

Author

Henry Bottomley, Jun 26 2000

Status

approved

A265609 Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n >= 0 and k >= 0.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 6, 0, 1, 4, 12, 24, 24, 0, 1, 5, 20, 60, 120, 120, 0, 1, 6, 30, 120, 360, 720, 720, 0, 1, 7, 42, 210, 840, 2520, 5040, 5040, 0, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0

Offset

0,8

Comments

The Pochhammer function is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1.

From Antti Karttunen, Dec 19 2015: (Start)

Apart from the initial row of zeros, if we discard the leftmost column and divide the rest of terms A(n,k) with (n+k) [where k is now the once-decremented column index of the new, shifted position] we get the same array back. See the given recursive formula.

When the numbers in array are viewed in factorial base (A007623), certain repeating patterns can be discerned, at least in a few of the topmost rows. See comment in A001710 and arrays A265890, A265892. (End)

A(n,k) is the k-th moment (about 0) of a gamma (Erlang) distribution with shape parameter n and rate parameter 1. - Geoffrey Critzer, Dec 24 2018

References

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 355.

Links

Table of n, a(n) for n=0..54.

NIST Digital Library of Mathematical Functions, Pochhammer's Symbol

Index entries for sequences related to factorial numbers

Formula

A(n,k) = Gamma(n+k)/Gamma(n) for n > 0 and n^k for n=0.

A(n,k) = Sum_{j=0..k} n^j*S1(k,j), S1(n,k) the Stirling cycle numbers A132393(n,k).

A(n,k) = (k-1)!/(Sum_{j=0..k-1} (-1)^j*binomial(k-1, j)/(j+n)) for n >= 1, k >= 1.

A(n,k) = (n+k-1)*A(n,k-1) for k >= 1, A(n,0) = 1. - Antti Karttunen, Dec 19 2015

E.g.f. for row k: 1/(1-x)^k. - Geoffrey Critzer, Dec 24 2018

A(n, k) = FallingFactorial(n + k - 1, k). - Peter Luschny, Mar 22 2022

G.f. for row n as a continued fraction of Stieltjes type: 1/(1 - n*x/(1 - x/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 3*x/(1 - ... ))))))). See Wall, Chapter XVIII, equation 92.5. Cf. A226513. - Peter Bala, Aug 27 2023

Example

Square array A(n,k) [where n=row, k=column] is read by ascending antidiagonals as:
A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...
Array starts:
n\k [0  1   2    3     4      5        6         7          8]
--------------------------------------------------------------
[0] [1, 0,  0,   0,    0,     0,       0,        0,         0]
[1] [1, 1,  2,   6,   24,   120,     720,     5040,     40320]
[2] [1, 2,  6,  24,  120,   720,    5040,    40320,    362880]
[3] [1, 3, 12,  60,  360,  2520,   20160,   181440,   1814400]
[4] [1, 4, 20, 120,  840,  6720,   60480,   604800,   6652800]
[5] [1, 5, 30, 210, 1680, 15120,  151200,  1663200,  19958400]
[6] [1, 6, 42, 336, 3024, 30240,  332640,  3991680,  51891840]
[7] [1, 7, 56, 504, 5040, 55440,  665280,  8648640, 121080960]
[8] [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200]
.
Seen as a triangle, T(n, k) = Pochhammer(n - k, k), the first few rows are:
   [0] 1;
   [1] 1, 0;
   [2] 1, 1,  0;
   [3] 1, 2,  2,   0;
   [4] 1, 3,  6,   6,    0;
   [5] 1, 4, 12,  24,   24,    0;
   [6] 1, 5, 20,  60,  120,  120,     0;
   [7] 1, 6, 30, 120,  360,  720,   720,     0;
   [8] 1, 7, 42, 210,  840, 2520,  5040,  5040,     0;
   [9] 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0.

Maple

for n from 0 to 8 do seq(pochhammer(n, k), k=0..8) od;

Mathematica

Table[Pochhammer[n, k], {n, 0, 8}, {k, 0, 8}]

Prog

(Sage)
for n in (0..8): print([rising_factorial(n, k) for k in (0..8)])
(Scheme)
(define (A265609 n) (A265609bi (A025581 n) (A002262 n)))
(define (A265609bi row col) (if (zero? col) 1 (* (+ row col -1) (A265609bi row (- col 1)))))
;; Antti Karttunen, Dec 19 2015

Crossrefs

Triangle giving terms only up to column k=n: A124320.

Row 0: A000007, row 1: A000142, row 3: A001710 (from k=1 onward, shifted two terms left).

Column 0: A000012, column 1: A001477, column 2: A002378, columns 3-7: A007531, A052762, A052787, A053625, A159083 (shifted 2 .. 6 terms left respectively, i.e. without the extra initial zeros), column 8: A239035.

Row sums of the triangle: A000522.

A(n, n) = A000407(n-1) for n>0.

2^n*A(1/2,n) = A001147(n).

Cf. also A007623, A008279 (falling factorial), A173333, A257505, A265890, A265892.

Keyword

nonn,tabl

Author

Peter Luschny, Dec 19 2015

Status

approved

A174295 Matrix inverse of A174294.

1, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 1, -2, -1, 1, 0, 1, -3, 2, 0, -2, 2, 0, 1, 6, -7, -3, 3, -3, 3, 0, 1, -15, 14, 3, -10, 7, -4, 4, 0, 1, 36, -37, -12, 19, -19, 12, -5, 5, 0, 1, -91, 90, 24, -54, 42, -30, 18, -6, 6, 0, 1, 232, -233, -67, 127, -115, 73, -43, 25, -7, 7

Offset

0,17

Comments

First column is a signed version of A099323 with an additional leading 1.

First 5 rows as in A054525.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

Sum_{k=0..n} T(n, k) = A000007(n).

T(n, 0) = A174297(n).

Example

Table begins:
  n\k|...0...1...2...3...4...5...6...7...8...9..10
  ---|--------------------------------------------
  0..|...1
  1..|..-1...1
  2..|..-1...0...1
  3..|...0..-1...0...1
  4..|..-1...0...0...0...1
  5..|...1..-2..-1...1...0...1
  6..|..-3...2...0..-2...2...0...1
  7..|...6..-7..-3...3..-3...3...0...1
  8..|.-15..14...3.-10...7..-4...4...0...1
  9..|..36.-37.-12..19.-19..12..-5...5...0...1
  10.|.-91..90..24.-54..42.-30..18..-6...6...0...1

Mathematica

t[n_, k_]:= t[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, If[k==1, Mod[n, 2], t[n-1, k-1] +t[n-2, k-1] -t[n-1, k] -t[n-2, k] ]]]; (* t = A174294 *)
M:= With[{m=30}, Table[t[n, k], {n, 0, m}, {k, 0, m}]];
T:= Inverse[M];
Table[T[[n+1, k+1]], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 25 2021 *)

Crossrefs

Cf. A000007 (row sums), A054525, A099323, A112467, A112468, A174294, A174296, A174297.

Keyword

sign,tabl

Author

Mats Granvik, Mar 15 2010

Status

approved

A115353 The mode of the digits of n (using smallest mode if multimodal).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 0, 1, 2, 3, 4, 5, 6, 6, 6, 6, 0, 1, 2, 3, 4, 5, 6, 7, 7, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 0, 0, 0

Offset

0,3

Comments

a(101)=1 and A054054(101)=0, but all previous terms are equivalent.

Links

Bence Bernáth, Table of n, a(n) for n = 0..10000

Example

a(12)=1 because 1, 2, the digits of 12, each occur the same number of times and 1 is the smaller of the two modes.
a(101)=1 because 1 is the unique mode of 1, 0, 1 (occurring twice while 0 appears only once).

Mathematica

a[n_] := Min[Commonest[IntegerDigits[n]]]; Array[a, 105, 0] (* Stefano Spezia, Jan 08 2023 *)

Prog

(MATLAB)
function nth_term=A115353(n)
     nth_term=mode((num2str(n)-'0'));
end
sequence = arrayfun(@A115353, linspace(0, 105, 106))
% Bence Bernáth, Jan 06 2023
(Python)
from statistics import mode
def a(n): return int(mode(sorted(str(n))))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jan 08 2023

Crossrefs

Cf. A054054 (Smallest digit of n).

Keyword

base,nonn

Author

Rick L. Shepherd, Jan 21 2006

Status

approved