cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)*binomial(-k,j-1)*x^j/j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 13, 24, 1, 1, 1, 5, 22, 73, 120, 1, 1, 1, 6, 33, 154, 501, 720, 1, 1, 1, 7, 46, 273, 1306, 4051, 5040, 1, 1, 1, 8, 61, 436, 2721, 12976, 37633, 40320, 1, 1, 1, 9, 78, 649, 4956, 31701, 147484, 394353, 362880, 1
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2017

Keywords

Examples

			Square array B(j,k) begins:
   1,   1,   1,    1,    1, ...
   0,   1,   2,    3,    4, ...
   0,   1,   3,    6,   10, ...
   0,   1,   4,   10,   20, ...
   0,   1,   5,   15,   35, ...
   0,   1,   6,   21,   56, ...
Square array A(n,k) begins:
   1,   1,   1,    1,    1, ...
   1,   1,   1,    1,    1, ...
   1,   2,   3,    4,    5, ...
   1,   6,  13,   22,   33, ...
   1,  24,  73,  154,  273, ...
   1, 120, 501, 1306, 2721, ...

Links

Crossrefs

Columns k=0..10 give A000012, A000142, A000262, A049376, A049377, A049378, A049402, A132164, A293986, A293987, A293988.
Rows n=0-1 give A000012.
Main diagonal gives A293989.
Cf. A293012, A293991.

Programs

  • Mathematica
    B[j_, k_] := (-1)^(j-1)*Binomial[-k, j-1];
A[0, ] = 1; A[n, k_] := (n-1)!*Sum[B[j, k]*A[n-j, k]/(n-j)!, {j, 1, n}];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

Formula

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

A049352 A triangle of numbers related to triangle A030524.

Original entry on oeis.org

1, 4, 1, 20, 12, 1, 120, 128, 24, 1, 840, 1400, 440, 40, 1, 6720, 16240, 7560, 1120, 60, 1, 60480, 201600, 129640, 27720, 2380, 84, 1, 604800, 2681280, 2275840, 656320, 80080, 4480, 112, 1, 6652800, 38142720, 41370560, 15402240, 2498160, 196560
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,1) = A001715(n+2). a(n,m)=: S1p(4; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers), S1p(3; n,m)= A046089(n,m).
The signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035469(n,m) := S2(4; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+3 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001715. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

Examples

			Triangle starts:
{1};
{4,1};
{20,12,1};
{120,128,24,1};
{840,1400,440,40,1};
...
E.g. Row polynomial E(3,x)= 20*x + 12*x^2 + x^3.
a(4,2)=128=4*(4*5)+3*(4*4) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*4*5)=20 colored versions, e.g. ((1c1),(2c1,3c4,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 4 colors, c1..c4, can be chosen and the vertex labeled 4 with j=2 can come in 5 colors, e.g. c1..c5. Therefore there are 4*((1)*(1*4*5))=80 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*4)*(1*4))=48 such forests, e.g. ((1c1,3c2)(2c1,4c4)) or ((1c1,3c3)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

Links

Crossrefs

Cf. A049377 (row sums).
Alternating row sums A134137.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+3)!/6, 10); # Peter Luschny, Jan 28 2016
  • Mathematica
    a[n_, k_] := (n!* Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n+3*j-1, 3*j-1], {j, 1, k}])/(3^k*k!); Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(# + 3)!/6&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
  • Maxima
    a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+3*j-1,3*j-1),j,1,k))/(3^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

Formula

a(n, m) = n!*A030524(n, m)/(m!*3^(n-m)); a(n, m) = (3*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n
a(n,k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+3*j-1,3*j-1)))/(3^k*k!). [Vladimir Kruchinin, Apr 01 2011]

A157384 A partition product of Stirling_1 type [parameter k = -4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 20, 1, 72, 80, 120, 1, 280, 1000, 600, 840, 1, 1740, 9200, 9000, 5040, 6720, 1, 8484, 79100, 138600, 88200, 47040, 60480, 1, 57232, 874720, 1789200, 1552320, 940800, 483840, 604800, 1, 328752, 9532880
Offset: 1

Views

Author

Peter Luschny, Mar 07 2009, Mar 14 2009

Keywords

Comments

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144354.
Same partition product with length statistic is A049352.
Diagonal a(A000217(n)) = rising_factorial(4,n-1), A001715(n+2).
Row sum is A049377.

Links

Crossrefs

Cf. A157386, A157384, A157383, A157400, A126074, A157391, A157392, A157393, A157394, A157395

Formula

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-2).

A144354 Partition number array, called M31(4), related to A049352(n,m)= |S1(4;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 4, 1, 20, 12, 1, 120, 80, 48, 24, 1, 840, 600, 800, 200, 240, 40, 1, 6720, 5040, 7200, 4000, 1800, 4800, 960, 400, 720, 60, 1, 60480, 47040, 70560, 84000, 17640, 50400, 28000, 33600, 4200, 16800, 6720, 700, 1680, 84, 1, 604800, 483840, 752640, 940800, 504000, 188160
Offset: 1

Views

Author

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Keywords

Comments

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(4;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
Fourth member (K=4) in the family M31(K) of partition number arrays.
If M31(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(4)|:= A049352.

Examples

			[1];[4,1];[20,12,1];[120,80,48,24,1];[840,600,800,200,240,40,1];...
a(4,3)= 48 = 3*|S1(4;2,1)|^2. The relevant partition of 4 is (2^2).

Links

Crossrefs

A049377 (row sums).
A144353 (M31(3) array), A144355 (M31(5) array).

Formula

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(4;j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)|= A001715(n+2) = (n+2)!/3!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. M3(n,k)=A036040.

A134138 Alternating row sums of triangle A046089 (S1p(3)).

Original entry on oeis.org

1, 2, 4, 2, -74, -916, -8672, -73564, -542852, -2595016, 18348496, 906083672, 21021502984, 406255974032, 7157641045696, 116383645516784, 1681549859135248, 18311613681506336, -3332917116147392
Offset: 1

Views

Author

Wolfdieter Lang Oct 12 2007

Keywords

Crossrefs

Cf. A049377 (row sums of A046089).

Programs

  • Mathematica
    Rest[CoefficientList[Series[1-E^(-x*(2-x)/(2*(1-x)^2)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 09 2013 *)

Formula

a(n) = Sum_{m=1..n} A046089(n,m)*(-1)^(m-1), n >= 1.
E.g.f.: 1 - exp(-x*(2-x)/(2*(1-x)^2)). Cf. e.g.f. first column of A046089.
a(n) = (3*n-4)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 09 2013
Lim sup n->infinity |a(n)|/(2*n^(n-1/6)*exp(-n^(1/3)/4+3*n^(2/3)/4-n+1/3)/sqrt(3)) = 1. - Vaclav Kotesovec, Oct 09 2013

A380259 Expansion of e.g.f. exp( (1/(1-2*x)^(3/2) - 1)/3 ).

Original entry on oeis.org

1, 1, 6, 51, 561, 7566, 120711, 2221311, 46269126, 1075249881, 27560477331, 771948530046, 23446574573841, 767288588019201, 26905482997736526, 1006166248423254171, 39962774633459923881, 1679677496419394133846, 74471142324541556576151
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2025

Keywords

Links

Crossrefs

Cf. A049377, A380212.

Programs

  • Mathematica
    Table[Sum[3^k * 2^(n-k) * Abs[StirlingS1[n,k]] * BellB[k, 1/3], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 23 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-2*x)^(3/2)-1)/3)))

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * |Stirling1(n,k)| * A004212(k) = Sum_{k=0..n} 3^k * 2^(n-k) * |Stirling1(n,k)| * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.
a(n) = (1/exp(1/3)) * (-2)^n * n! * Sum_{k>=0} binomial(-3*k/2,n)/(3^k * k!).
a(n) ~ 2^(n + 3/10) * n^(n - 1/5) * exp(-1/3 + 2^(1/5)*n^(1/5)/4 + 5*2^(3/5)*n^(3/5)/6 - n) / sqrt(5) * (1 + 2^(4/5) / (30 * n^(1/5))). - Vaclav Kotesovec, Jan 23 2025
Showing 1-6 of 6 results.

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