A293991 -id:A293991

A049425

Row sums of triangle A049404.

+10
9

1, 1, 3, 9, 33, 141, 651, 3333, 18369, 108153, 678771, 4495041, 31324833, 228803589, 1744475643, 13852095741, 114235118721, 976176336753, 8627940414819, 78726234866553, 740440277799201, 7168107030092541, 71331617341611243, 728811735008913909, 7637128289949856833, 81995144342947130601

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..616 (terms 0..200 from Vincenzo Librandi)

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, J. Integer Seqs., Vol. 20 (2017), #17.8.2.

FORMULA

E.g.f.: exp(x+x^2+(x^3)/3).

a(n) = n! * sum(k=0..n, sum(j=0..k, binomial(3*j,n) * (-1)^(k-j)/(3^k * (k-j)!*j!))). [Vladimir Kruchinin, Feb 07 2011]

Conjecture: -a(n) +a(n-1) +(2*n-2)*a(n-2) + (2-3*n+n^2)*a(n-3)=0. - R. J. Mathar, Nov 14 2011

a(n) ~ exp(n^(2/3)+n^(1/3)/3-2*n/3-2/9)*n^(2*n/3)/sqrt(3)*(1+59/(162*n^(1/3))). - Vaclav Kotesovec, Oct 08 2012

From Emanuele Munarini, Oct 20 2014: (Start)

Recurrence: a(n+3) = a(n+2)+2*(n+2)*a(n+1)+(n+2)*(n+1)*a(n).

It derives from the differential equation for the e.g.f.: A'(x) = (1+2*x+x^2)*A(x).

So, the above conjecture is true.

b(n) = a(n+1) = sum((n!/k!)*sum(bin(k,i)*bin(k-i+2,n-2*i-k)/3^i,i=0..k),k=0..n).

E.g.f. for b(n) = a(n+1): (1+t)^2*exp(t+t^2+t^3/3).

(End)

a(n) = Sum_{k=0..n} Stirling1(n,k) * A004212(k). - Seiichi Manyama, Jan 31 2024

MATHEMATICA

Table[n!*SeriesCoefficient[E^(x+x^2+(x^3)/3), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)

PROG

(PARI) x='x+O('x^66); Vec(serlaplace(exp(x+x^2+(x^3)/3))) \\ Joerg Arndt, May 04 2013

(Maxima) /* for b(n) = a(n+1) */

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

makelist(b(n), n, 0, 24); /* Emanuele Munarini, Oct 20 2014 */

CROSSREFS

Column k=2 of A293991.

Cf. A004212.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

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).

+10
8

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

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

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.

EXAMPLE

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, 2, 3, 4, 5, ...

1, 6, 13, 22, 33, ...

1, 24, 73, 154, 273, ...

1, 120, 501, 1306, 2721, ...

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 *)

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.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Oct 21 2017

STATUS

approved

A049426

Row sums of triangle A049410.

+10
6

1, 1, 4, 16, 76, 436, 2776, 19384, 148576, 1226656, 10824256, 101695936, 1010783104, 10577428096, 116166090496, 1334409569536, 15985101216256, 199216504113664, 2577292524107776, 34542575915216896, 478781761481291776

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..547

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

FORMULA

E.g.f.: exp((-1+(1+x)^4)/4).

a(n) = n!*Sum_(k=1..n, Sum_(j=0..k, binomial(4*j,n)*(-1)^(k-j)/(4^k*(k-j)!*j!))). - Vladimir Kruchinin, Feb 07 2011

D-finite with recurrence a(n) -a(n-1) +3*(-n+1)*a(n-2) -3*(n-1)*(n-2)*a(n-3) -(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jun 23 2023

a(n) = Sum_{k=0..n} Stirling1(n,k) * A004213(k). - Seiichi Manyama, Jan 31 2024

MATHEMATICA

With[{nn=20}, CoefficientList[Series[Exp[((1+x)^4-1)/4], {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Jan 28 2017 *)

CROSSREFS

Cf. A004213, A049410.

Column of A293991.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

A049427

Row sums of triangle A049424.

+10
6

1, 1, 5, 25, 145, 1025, 8245, 72745, 704705, 7424065, 83940805, 1012504505, 12972555025, 175624847425, 2501468566325, 37364323364425, 583569693556225, 9504040277271425, 161021013457176325, 2832196631069755225, 51619359912771959825

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..510

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

FORMULA

E.g.f.: exp((-1+(1+x)^5)/5).

a(n) = n! * sum(k=0..n, sum(j=0..k, binomial(5*j,n) * (-1)^(k-j)/(5^k * (k-j)!*j!))). - Vladimir Kruchinin, Feb 07 2011

D-finite with recurrence a(n) -a(n-1) +4*(-n+1)*a(n-2) -6*(n-1)*(n-2)*a(n-3) -4*(n-1)*(n-2)*(n-3)*a(n-4) -(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 23 2023

a(n) = Sum_{k=0..n} Stirling1(n,k) * A005011(k). - Seiichi Manyama, Jan 31 2024

CROSSREFS

Column of A293991.

Row sums of A157394.

Cf. A005011.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

A049428

Row sums of triangle A049411.

+10
6

1, 1, 6, 36, 246, 2046, 19716, 209616, 2441916, 31050396, 425883816, 6244077456, 97391939976, 1609040166696, 28029696862896, 512903202039936, 9829166157390096, 196739739722616336, 4102788435212513376, 88945209649582514496, 2000700796384204930656

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..485

Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

FORMULA

E.g.f.: exp((-1+(1+x)^6)/6).

a(n) = n! * Sum_{k=1..n} Sum_{j=0..k} binomial(6*j,n) *(-1)^(k-j)/ (6^k*(k-j)!*j!). - Vladimir Kruchinin, Feb 07 2011

D-finite with recurrence a(n) -a(n-1) +5*(-n+1)*a(n-2) -10*(n-1)*(n-2)*a(n-3) -10*(n-1)*(n-2)*(n-3)*a(n-4) -5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)=0. - R. J. Mathar, Jun 23 2023

a(n) = Sum_{k=0..n} Stirling1(n,k) * A005012(k). - Seiichi Manyama, Jan 31 2024

MATHEMATICA

nmax = 20;

a[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, nmax}]];

a[0] = 1; a[n_] := Sum[a[n, m], {m, 1, n}];

Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 27 2018 *)

CROSSREFS

Column k=5 of A293991.

Cf. A005012.

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

EXTENSIONS

Offset adjusted by R. J. Mathar, Aug 29 2009

STATUS

approved

A294042

Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp((1+x)^k - 1).

+10
6

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 87, 76, 1, 0, 1, 6, 45, 232, 585, 312, 1, 0, 1, 7, 66, 485, 2248, 4383, 1384, 1, 0, 1, 8, 91, 876, 6145, 24544, 35919, 6512, 1, 0, 1, 9, 120, 1435, 13716, 88245, 295456, 318195, 32400, 1, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

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

FORMULA

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

A(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * Bell(j). - Seiichi Manyama, Jan 31 2024

EXAMPLE

Square array A(n,k) begins:

1, 1, 1, 1, 1, 1, ...

0, 1, 2, 3, 4, 5, ...

0, 1, 6, 15, 28, 45, ...

0, 1, 20, 87, 232, 485, ...

0, 1, 76, 585, 2248, 6145, ...

0, 1, 312, 4383, 24544, 88245, ...

CROSSREFS

Columns k=0..5 give A000007, A000012, A000898, A192989, A202824, A202825.

Rows n=0..2 give A000012, A001477, A000384.

Main diagonal gives A294045.

Cf. A000110, A293991.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Oct 22 2017

STATUS

approved

A294118

Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(k*((1+x)^k - 1)).

+10
4

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 9, 20, 1, 0, 1, 16, 99, 112, 1, 0, 1, 25, 304, 1233, 688, 1, 0, 1, 36, 725, 6496, 16929, 4544, 1, 0, 1, 49, 1476, 23425, 152416, 251829, 31936, 1, 0, 1, 64, 2695, 66816, 826225, 3867136, 4012011, 236800, 1, 0, 1, 81, 4544, 162337

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

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

FORMULA

A(0,k) = 1 and A(n,k) = k^2 * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1)*A(n-j,k)/(n-j)! for n > 0.

EXAMPLE

Square array A(n,k) begins:

1, 1, 1, 1, 1, ...

0, 1, 4, 9, 16, ...

0, 1, 20, 99, 304, ...

0, 1, 112, 1233, 6496, ...

0, 1, 688, 16929, 152416, ...

0, 1, 4544, 251829, 3867136, ...

CROSSREFS

Columns k=0..3 give A000007, A000012, A294119, A294120.

Rows n=0..1 give A000012, A000290.

Main diagonal gives A294191.

Cf. A293991, A294042.

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Oct 23 2017

STATUS

approved

A294003

a(n) = n! * [x^n] exp(((1+x)^(n+1) - 1)/(n+1)).

+10
3

1, 1, 3, 16, 145, 2046, 40831, 1078288, 36229761, 1508861980, 76260702331, 4591005777216, 323906453871793, 26419909067594056, 2463658914613015575, 260204512488337862656, 30878562368653933102081, 4088281211853327516484368, 600094288103991706472584051

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..281

CROSSREFS

Main diagonal of A293991.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 21 2017

STATUS

approved