A188062 -id:A188062 - cp's OEIS frontend

A188066 Triangle read by rows: Bell polynomial of the second kind B(n,k) with argument vector (7, 42, 210, 840, 2520, 5040, 5040).

7, 42, 49, 210, 882, 343, 840, 11172, 12348, 2401, 2520, 117600, 288120, 144060, 16807, 5040, 1076040, 5433120, 5330220, 1512630, 117649, 5040, 8643600, 89029080, 155296680, 81177810, 14823774, 823543, 0, 60540480, 1306912320, 3884433840, 3360055440, 1087076760, 138355224, 5764801

Offset: 1

Vladimir Kruchinin, Mar 24 2011

From the explicit write-up of the Bell polynomials we have B(n,k)(7*x^6, 42*x^5, 210*x^4, 840*x^3, 2520*x^2, 5040*x, 5040) = B(n,k)(7, 42, ..., 5040)*x^(7*k-n) for a more general set of arguments.

			Triangle begins
     7;
    42,      49;
   210,     882,     343;
   840,   11172,   12348,    2401;
  2520,  117600,  288120,  144060,   16807;
  5040, 1076040, 5433120, 5330220, 1512630, 117649;
  ...

M. Abbas and S. Bouroubi, On new identities for Bell's polynomials, Disc. Math. 293 (1-3) (2005), 5-10.
Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2011.
John Riordan, Derivatives of composite functions, Bull. Am. Math. Soc. 52 (1946), 664-667.
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A188062, A068424 (row 7).

A188066 := proc(n,k) n!/k!*add( binomial(k,j)*binomial(7*j,n)*(-1)^(k-j),j=0..k) ; end proc:
seq(seq(A188066(n,k),k=1..n),n=1..5) ; # R. J. Mathar, Apr 08 2011
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n<7,[7,42,210,840,2520,5040,5040][n+1],0), 9); # Peter Luschny, Jan 29 2016

b[n_, k_] := n!/k!*Sum[ Binomial[k, j]*Binomial[7*j, n]*(-1)^(k - j), {j, 0, k}]; Table[b[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 21 2013, translated from Maxima *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, If[n<7, {7, 42, 210, 840, 2520, 5040, 5040}[[n + 1]], 0]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

B(n,k):=n!/k!*x^(7*k-n)*sum(binomial(k,j)*binomial(7*j,n)*(-1)^(k-j),j,0,k);

B(n,k) = (n!/k!)*Sum_{j=0..k} binomial(k,j)*binomial(7*j,n)*(-1)^(k-j).

A187084 Exponential Riordan array (1,4x+6x^2+4*x^3+x^4).

Original entry on oeis.org

4, 12, 16, 24, 144, 64, 24, 816, 1152, 256, 0, 3360, 12480, 7680, 1024, 0, 10080, 100800, 134400, 46080, 4096, 0, 20160, 645120, 1747200, 1182720, 258048, 16384, 0, 20160, 3306240, 18305280, 22364160, 9117696, 1376256, 65536, 0, 0, 13305600, 159667200, 341591040, 235339776, 63995904, 7077888, 262144

Offset: 1

Vladimir Kruchinin, Mar 04 2011

The column of index 0 contains a 1 followed by zeros and is not incorporated here.
Also the Bell polynomials of the second kind B(n,k)(4,12,24,24).
If the argument vector is generalized to contain falling powers of a variable x, B(n,k)(4*x^3,12*x,24*x,24) =sum_{j=0..k} binomial(k,j) *sum_{i=j..n-k+j} 6^(i-j) *binomial(j,i-j) *binomial(k-j,n-3*k+3*j-i) *4^(4*k-n-2*j) *x^(4*k-n) *n!/k!.

			Triangle begins:
  4;
  12,16;
  24,144,64;
  24,816,1152,256;
  0,3360,12480,7680,1024;
  0,10080,100800,134400,46080,4096;
  0,20160,645120,1747200,1182720,258048,16384;
  0,20160,3306240,18305280,22364160,9117696,1376256,65536;
  ...

M. Abbas and S. Bouroubi, On new identities for Bell's polynomials, Disc. Math 293. (1-3) (2005) 5-10
Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2011.
John Riordan, Derivatives of composite functions, Bull. Am. Math. Soc. 52 (1946) 664
Eric W. Weisstein, Bell Polynomial, MathWorld

Cf. A008279 (its row 4), A188138, A188062, A188066.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n<4,[4,12,24,24][n+1],0), 9); # Peter Luschny, Jan 29 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, If[n<4, {4, 12, 24, 24}[[n+1]], 0]], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

B(n,k):=n!/k!*sum(binomial(k,j)*sum(6^(i-j)*binomial(j,i-j)*binomial(k-j,n-3*k+3*j-i),i,j,n-k+j)*4^(4*k-n-2*j),j,0,k);

B(n,k) = (n!/k!)*Sum_{j=0..k} binomial(k,j)*( Sum_{i=j..n-k+j} 6^(i-j)*binomial(j,i-j)*binomial(k-j,n-3*k+3*j-i) )*4^(4*k-n-2*j).

cp's OEIS Frontend

A188066 Triangle read by rows: Bell polynomial of the second kind B(n,k) with argument vector (7, 42, 210, 840, 2520, 5040, 5040).

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A187084 Exponential Riordan array (1,4x+6x^2+4*x^3+x^4).

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cp's OEIS Frontend

A188066 Triangle read by rows: Bell polynomial of the second kind B(n,k) with argument vector (7, 42, 210, 840, 2520, 5040, 5040).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A187084 Exponential Riordan array (1,4*x+6*x^2+4*x^3+x^4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A187084 Exponential Riordan array (1,4x+6x^2+4*x^3+x^4).