Nelma Moreira - cp's OEIS frontend

User: Nelma Moreira

Nelma Moreira has authored 4 sequences.

A099266 Partial sums of A056273.

1, 3, 8, 23, 75, 278, 1154, 5265, 25913, 135212, 736704, 4139831, 23767895, 138468210, 814675838, 4824766301, 28699128501, 171207852152, 1023332115836, 6124430348355, 36684624841811, 219860794899518, 1318179574171578

Offset: 1

Nelma Moreira, Oct 10 2004

Some previous names were a(6,n) := (1/600)*6^n + (1/36)*4^n + (1/12)*3^n + (3/8)*2^n + (11/30)*n - (439/900) = Sum_{m=1..n} Sum_{i=1..6} S(m,i), where S(n,i) = A008277(n,i) are the Stirling numbers of the second kind.

Density of the regular language L{0}* over {0, 1, 2, 3, 4, 5, 6} (i.e., the number of strings of length n), where L is described by regular expression with c = 6: Sum_{i=1..c} Prod_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation. I.e., L = L((11* + ... + 11*2(1 + 2)*3(1 + 2 + 3)*4(1 + 2 + 3 + 4)*5(1 + 2 + 3 + 4 + 5)*6(1 + 2 + 3 + 4 + 5 + 6)*)0*).

Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC & LIACC, Universidade do Porto.
Nelma Moreira and Rogerio Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Index entries for linear recurrences with constant coefficients, signature (17,-111,355,-584,468,-144).

Cf. A047926, A056273, A008277, A099264, A099265.

with (combinat):seq(sum(sum(stirling2(k, j),j=1..6), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007

Vec(x*(91*x^4-135*x^3+68*x^2-14*x+1)/((x-1)^2*(2*x-1)*(3*x-1)*(4*x-1)*(6*x-1)) + O(x^100)) \\ Colin Barker, Oct 28 2014

a(n) = sum(m=1, n, sum(i=1, 6, stirling(m, i, 2))) \\ Petros Hadjicostas, Mar 09 2021

For c = 6, a(c, n) = g(1, c)*n + Sum_{k=2..c} g(k, c)*k*(k^n - 1)/(k - 1), where g(1, 1) = 1, g(1, c) = g(1, c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k, c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.

G.f.: x*(91*x^4 - 135*x^3 + 68*x^2 - 14*x + 1) / ((x - 1)^2*(2*x - 1)*(3*x - 1)*(4*x - 1)*(6*x - 1)). - Colin Barker, Oct 28 2014

Shorter name by Joerg Arndt, Oct 28 2014

Comments edited by Petros Hadjicostas, Mar 09 2021

A099265 Partial sums of A056272.

Original entry on oeis.org

1, 3, 8, 23, 75, 277, 1132, 4977, 22979, 109451, 531456, 2610931, 12917683, 64181625, 319695980, 1594859885, 7963472187, 39784944799, 198827606704, 993846943839, 4968361974491, 24839192686973, 124188113975628, 620917025694793, 3104514504312595, 15522360665856147, 77611167795714752

Offset: 1

Nelma Moreira, Oct 10 2004

Density of regular language L{0}* over {0, 1, 2, 3, 4, 5} (i.e., the number of strings of length n), where L is described by regular expression with c = 5: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation. I.e., L = L((11* + 11*2(1 + 2)* + ... + 11*2(1 + 2)*3(1 + 2 + 3)*4(1 + 2 + 3 + 4)*5(1 + 2 + 3 + 4 + 5)*)0*).

Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC & LIACC, Universidade do Porto.
Nelma Moreira and Rogerio Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.

Cf. A008277, A047926, A056272, A099264, A099266.

with (combinat):seq(sum(sum(stirling2(k, j),j=1..5), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007

a(n) = sum(m=1, n, sum(i=1, 5, stirling(m, i, 2))) \\ Petros Hadjicostas, Mar 10 2021

a(5,n) = (1/96)*5^n + (1/8)*3^n + (1/3)*2^n + (3/8)*n - 15/32.
a(n) = Sum_{m=1..n} Sum_{i=1..5} S(m,i), where S(m,i) = A008277(m,i) (i.e., partial sum of the sum of Stirling numbers of second kind S(n,i) for i = 1..5).
For c = 5, a(c,n) = g(1,c)*n + Sum_{k=2..c} g(k,c)*k*(k^n - 1)/(k - 1), where g(1,1) = 1, g(1,c) = g(1,c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k,c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.
G.f.: x*(-1 + 19*x^3 - 24*x^2 + 9*x)/((3*x-1)*(2*x-1)*(5*x-1)*(x-1)^2). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009]

Name and Formula section edited by Petros Hadjicostas, Mar 10 2021

More terms from Michel Marcus, Jan 05 2025

A099263 a(n) = (1/40320)8^n + (1/1440)6^n + (1/360)5^n + (1/64)4^n + (11/180)3^n + (53/288)2^n + 103/280. Partial sum of Stirling numbers of second kind S(n,i), i=1..8 (i.e., a(n) = Sum_{i=1..8} S(n,i)).

Original entry on oeis.org

1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115929, 677359, 4189550, 27243100, 184941915, 1301576801, 9433737120, 69998462014, 529007272061, 4054799902003, 31415584940850, 245382167055488, 1928337630016767, 15222915798289765, 120582710957928740, 957566218595705122, 7618489083072350433

Offset: 1

Nelma Moreira, Oct 10 2004

Density of regular language L over {1,2,3,4,5,6,7,8} (i.e., number of strings of length n in L) described by a regular expression with c = 8: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
N. Moreira and R. Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.
N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Index entries for linear recurrences with constant coefficients, signature (29,-343,2135,-7504,14756,-14832,5760).

Cf. A007051, A007581, A056272, A056273, A099262.
A row of the array in A278984.
Cf. A008277 (Stirling2), A248925.

[(1/40320)*8^n+(1/1440)*6^n+(1/360)*5^n+(1/64)*4^n +(11/180)*3^n+(53/288)*2^n+103/280: n in [1..30]]; // Vincenzo Librandi, Jul 27 2017

CoefficientList[Series[-(3641 x^6 - 6583 x^5 + 4566 x^4 - 1579 x^3 + 290 x^2 - 27 x + 1) / ((x - 1) (2 x - 1) (3 x - 1) (4 x - 1) (5 x - 1) (6 x - 1) (8 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 27 2017 *)
Table[Sum[StirlingS2[n, k], {k, 0, 8}], {n, 1, 30}] (* Robert A. Russell, Apr 25 2018 *)
LinearRecurrence[{29,-343,2135,-7504,14756,-14832,5760},{1,2,5,15,52,203,877},30] (* Harvey P. Dale, Aug 27 2019 *)

a(n) = (1/40320)*8^n + (1/1440)*6^n + (1/360)*5^n + (1/64)*4^n + (11/180)*3^n + (53/288)*2^n + 103/280; \\ Altug Alkan, Apr 25 2018

For c = 8, a(n) = c^n/c! + Sum_{k=1..c-2} k^n/k! * Sum_{j=2..c-k} (-1)^j/j!, or = Sum_{k=1..c} g(k, c)*k^n, where g(1, 1) = 1, g(1, c) = g(1, c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k, c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.
G.f.: -x*(3641*x^6 - 6583*x^5 + 4566*x^4 - 1579*x^3 + 290*x^2 - 27*x + 1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). [Colin Barker, Dec 05 2012]
a(n) = Sum_{k=0..8} Stirling2(n,k).
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} (1 - j*x) with k = 8. - Robert A. Russell, Apr 25 2018

More terms from Michel Marcus, Jan 05 2025

A099262 a(n) = (1/5040)7^n + (1/240)5^n + (1/72)4^n + (1/16)3^n + (11/60)*2^n + 53/144. Partial sum of Stirling numbers of second kind S(n,i), i=1..7 (i.e., a(n) = Sum_{i=1..7} S(n,i)).

Original entry on oeis.org

1, 2, 5, 15, 52, 203, 877, 4139, 21110, 115179, 665479, 4030523, 25343488, 164029595, 1084948961, 7291973067, 49582466986, 339971207051, 2345048898523, 16244652278171, 112871151708404, 785938550025147, 5480960778389365, 38264428799608235, 267342497477336542, 1868866831126685483

Offset: 1

Nelma Moreira, Oct 10 2004

Density of regular language L over {1,2,3,4,5,6,7} (i.e., number of strings of length n in L) described by regular expression with c=7: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*) where Sum stands for union and Product for concatenation.

Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
N. Moreira and R. Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.
Index entries for linear recurrences with constant coefficients, signature (22,-190,820,-1849,2038,-840).

Cf. A007051, A007581, A056272, A056273, A099263.

A row of the array in A278984.

Table[Sum[StirlingS2[n, k], {k, 0, 7}], {n, 1, 30}] (* Robert A. Russell, Apr 25 2018 *)

a(n) = (1/5040)*7^n + (1/240)*5^n + (1/72)*4^n + (1/16)*3^n + (11/60)*2^n + 53/144; \\ Altug Alkan, Apr 25 2018

For c=7, a(n) = (c^n)/c! + Sum_{k=1..c-2} ((k^n)/k!*(Sum_{j=2..c-k}(((-1)^j)/j!))) or = Sum_{k=1..c} (g(k, c)*k^n) where g(1, 1)=1, g(1, c) = g(1, c-1)+((-1)^(c-1))/(c-1)!, c > 1, g(k, c) = g(k-1, c-1)/k, for c > 1 and 2 <= k <= c.
G.f.: -x*(531*x^5-881*x^4+535*x^3-151*x^2+20*x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)). - Colin Barker, Dec 05 2012
a(n) = Sum_{k=0..7} Stirling2(n,k).
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=7. - Robert A. Russell, Apr 25 2018

More terms from Michel Marcus, Jan 05 2025

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User: Nelma Moreira

A099266 Partial sums of A056273.

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A099265 Partial sums of A056272.

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A099263 a(n) = (1/40320)*8^n + (1/1440)*6^n + (1/360)*5^n + (1/64)*4^n + (11/180)*3^n + (53/288)*2^n + 103/280. Partial sum of Stirling numbers of second kind S(n,i), i=1..8 (i.e., a(n) = Sum_{i=1..8} S(n,i)).

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A099262 a(n) = (1/5040)*7^n + (1/240)*5^n + (1/72)*4^n + (1/16)*3^n + (11/60)*2^n + 53/144. Partial sum of Stirling numbers of second kind S(n,i), i=1..7 (i.e., a(n) = Sum_{i=1..7} S(n,i)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A099263 a(n) = (1/40320)8^n + (1/1440)6^n + (1/360)5^n + (1/64)4^n + (11/180)3^n + (53/288)2^n + 103/280. Partial sum of Stirling numbers of second kind S(n,i), i=1..8 (i.e., a(n) = Sum_{i=1..8} S(n,i)).

A099262 a(n) = (1/5040)7^n + (1/240)5^n + (1/72)4^n + (1/16)3^n + (11/60)*2^n + 53/144. Partial sum of Stirling numbers of second kind S(n,i), i=1..7 (i.e., a(n) = Sum_{i=1..7} S(n,i)).