A215862 -id:A215862 - cp's OEIS frontend

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A215861 Number T(n,k) of simple labeled graphs on n nodes with exactly k connected components that are trees or cycles; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 4, 3, 1, 0, 19, 19, 6, 1, 0, 137, 135, 55, 10, 1, 0, 1356, 1267, 540, 125, 15, 1, 0, 17167, 15029, 6412, 1610, 245, 21, 1, 0, 264664, 218627, 90734, 23597, 3990, 434, 28, 1, 0, 4803129, 3783582, 1515097, 394506, 70707, 8694, 714, 36, 1

Offset: 0

Alois P. Heinz, Aug 25 2012

Also the Bell transform of A215851(n+1). For the definition of the Bell transform see A264428 and the links given there. - Peter Luschny, Jan 21 2016

			T(4,2) = 19:
  .1 2.  .1 2.  .1-2.  .1-2.  .1 2.  .1 2.  .1 2.  .1 2.  .1 2.  .1 2.
  . /|.  .|\ .  .|/ .  . \|.  . /|.  .  |.  . / .  .|\ .  . \ .  .|  .
  .4-3.  .4-3.  .4 3.  .4 3.  .4 3.  .4-3.  .4-3.  .4 3.  .4-3.  .4-3.
  .
  .1-2.  .1-2.  .1 2.  .1-2.  .1-2.  .1 2.  .1-2.  .1 2.  .1 2.
  .|  .  . / .  .|/ .  . \ .  .  |.  . \|.  .   .  .| |.  . X .
  .4 3.  .4 3.  .4 3.  .4 3.  .4 3.  .4 3.  .4-3.  .4 3.  .4 3.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,     1;
  0,     4,     3,    1;
  0,    19,    19,    6,    1;
  0,   137,   135,   55,   10,   1;
  0,  1356,  1267,  540,  125,  15,   1;
  0, 17167, 15029, 6412, 1610, 245,  21,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A215851, A215852, A215853, A215854, A215855, A215856, A215857, A215858, A215859, A215860.
Diagonal and lower diagonals give: A000012, A000217, A215862, A215863, A215864, A215865.
Row sums give: A144164.
T(2n,n) gives A309313.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, add(binomial(n-1, i)*T(n-1-i, k-1)*
      `if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);
# Alternatively, with the function BellMatrix defined in A264428:
BellMatrix(n -> `if`(n<2, 1, n!/2+(n+1)^(n-1)), 8); # Peter Luschny, Jan 21 2016

t[0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0; t[n_, k_] := t[n, k] =Sum[ Binomial[n-1, i]*t[n-1-i, k-1]* If[i < 2, 1, i!/2 + (i+1)^(i-1)], {i, 0, n-k}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 07 2013 *)
(* Alternatively, with the function BellMatrix defined in A264428: *)
g[n_] =  If[n < 2, 1, n!/2 + (n+1)^(n-1)]; BellMatrix[g, 8] (* Peter Luschny, Jan 21 2016 *)
rows = 11;
t = Table[If[n<2, 1, n!/2 + (n+1)^(n-1)], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: factorial(n)//2 + (n+1)^(n-1) if n>=2 else 1, 8) # Peter Luschny, Jan 21 2016

T(0,0) = 1, T(n,k) = 0 for k<0 or k>n, and otherwise T(n,k) = Sum_{i=0..n-k} C(n-1,i)*T(n-1-i,k-1)*h(i) with h(i) = 1 for i<2 and h(i) = i!/2 + (i+1)^(i-1) else.

A241765 a(n) = n(n + 1)(n + 2)(3n + 17)/24.

Original entry on oeis.org

0, 5, 23, 65, 145, 280, 490, 798, 1230, 1815, 2585, 3575, 4823, 6370, 8260, 10540, 13260, 16473, 20235, 24605, 29645, 35420, 41998, 49450, 57850, 67275, 77805, 89523, 102515, 116870, 132680, 150040, 169048, 189805, 212415, 236985, 263625, 292448

Offset: 0

Bruno Berselli, Apr 28 2014

Equivalently, Sum_{i=0..n} (i+4)*A000217(i).
Sequences of the type Sum_{i=0..n} (i+k)*A000217(i):
k = 0,  A001296: 0,  1,  7, 25,  65, 140, 266, 462, ...
k = 1,  A000914: 0,  2, 11, 35,  85, 175, 322, 546, ...
k = 2,  A050534: 0,  3, 15, 45, 105, 210, 378, 630, ... (deleting two 0)
k = 3,  A215862: 0,  4, 19, 55, 125, 245, 434, 714, ...
k = 4,     a(n): 0,  5, 23, 65, 145, 280, 490, 798, ...
k = 5,  A239568: 0,  6, 27, 75, 165, 315, 546, 882, ...
Antidiagonal sums (without 0) give A034263: 1, 9, 39, 119, 294, ...
Diagonal: 1, 11, 45, 125, 280, 546, ... is A051740.
Also: k = -1 gives A050534 deleting a 0; k = -2 gives 0 followed by A059302.
After 0, partial sums of A212343 and third column of A118788.
This sequence is even related to A005286 by a(n) = n*A005286(n) - Sum_{i=0..n-1} A005286(i).

			a(7) = 4*0 + 5*1 + 6*3 + 7*6 + 8*10 + 9*15 + 10*21 + 11*28 = 798.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A005286, A118788, A212343, A227342.

Cf. similar sequences A000914, A001296, A050534, A059302, A215862, A239568 (see table in Comments lines).

/* By first comment: */ k:=4; A000217:=func; [&+[(i+k)*A000217(i): i in [0..n]]: n in [0..40]];

Maple

A241765:=n->n*(n + 1)*(n + 2)*(3*n + 17)/24; seq(A241765(n), n=0..40); # Wesley Ivan Hurt, May 09 2014

Mathematica

Table[n (n + 1) (n + 2) (3 n + 17)/24, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 23, 65, 145}, 40] CoefficientList[Series[x (5 - 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)

Maxima

makelist(coeff(taylor(x*(5-2*x)/(1-x)^5, x, 0, n), x, n), n, 0, 40);

PARI

a(n)=n*(n+1)*(n+2)*(3*n+17)/24 \\ Charles R Greathouse IV, Oct 07 2015

PARI

x='x+O('x^99); concat(0, Vec(x*(5-2*x)/(1-x)^5)) \\ Altug Alkan, Apr 10 2016

Sage

[n*(n+1)*(n+2)*(3*n+17)/24 for n in (0..40)]

Formula

G.f.: x*(5 - 2*x)/(1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = A227342(A055998(n+1)).

a(n) = Sum_{j=0..n+2} (-1)^(n-j)*binomial(-j,-n-2)*S1(j,n), S1 Stirling cycle numbers A132393. - Peter Luschny, Apr 10 2016

A354794 Triangle read by rows. The Bell transform of the sequence {m^m | m >= 0}.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 3, 1, 0, 27, 19, 6, 1, 0, 256, 175, 55, 10, 1, 0, 3125, 2101, 660, 125, 15, 1, 0, 46656, 31031, 9751, 1890, 245, 21, 1, 0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1, 0, 16777216, 11012415, 3463615, 688506, 95781, 9702, 714, 36, 1

Offset: 0

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Author

Peter Luschny, Jun 09 2022

Keywords

nonn
tabl

Comments

For the definition of the Bell transform see A264428. The Bell transform of {(-m)^m | m >= 0} is A039621. The numbers A039621(n, k) are known as the Lehmer-Comtet numbers of 2nd kind. We think it is more natural to use Bell_{n, k}({m^m}) as the basis for the definition (and let the triangle start at (0, 0)).

Examples

Triangle T(n, k) begins: [0] 1; [1] 0, 1; [2] 0, 1, 1; [3] 0, 4, 3, 1; [4] 0, 27, 19, 6, 1; [5] 0, 256, 175, 55, 10, 1; [6] 0, 3125, 2101, 660, 125, 15, 1; [7] 0, 46656, 31031, 9751, 1890, 245, 21, 1; [8] 0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1; [9] 0, 16777216, 11012415, 3463615, 688506, 95781, 9702, 714, 36, 1;

References

Louis Comtet, Advanced Combinatorics. Reidel, Dordrecht, 1974, p. 139-140.

Links

D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

Peter Luschny, The Bell transform

Wikipedia, Bell polynomials.

Crossrefs

Cf. A264428, A039621 (signed variant), A195979 (row sums), A000312 (column 1), A045531 (column 2), A281596 (column 3), A281595 (column 4), A000217 (diagonal 1), A215862 (diagonal 2), A354795 (matrix inverse), A137452 (Abel).

Programs

Maple

T := (n, k) -> if n = k then 1 else add((-1)^j*(n-j-1)^(n-1)/(j!*(k-1-j)!), j = 0.. k-1) fi: seq(seq(T(n, k), k = 0..n), n = 0..9); # Alternatively, using the function BellMatrix from A264428: BellMatrix(n -> n^n, 9); # Or by recursion: R := proc(n, k, m) option remember; if k < 0 or n < 0 then 0 elif k = 0 then 1 else m*R(n, k-1, m) + R(n-1, k, m+1) fi end: A039621 := (n, k) -> ifelse(n = 0, 1, R(k-1, n-k, n-k)):

Mathematica

Unprotect[Power]; Power[0, 0] = 1; pow[n_] := n^n; R = Range[0, 9]; T[n_, k_] := BellY[n, k, pow[R]]; Table[T[n, k], {n, R}, {k, 0, n}] // Flatten

Python

from functools import cache @cache def t(n, k, m): if k < 0 or n < 0: return 0 if k == 0: return n ** k return m * t(n, k - 1, m) + t(n - 1, k, m + 1) def A354794(n, k): return t(k - 1, n - k, n - k) if n != k else 1 for n in range(9): print([A354794(n, k) for k in range(n + 1)])

Formula

T(n, k) = Bell_{n, k}(A000312), where Bell_{n, k} is the partial Bell polynomial evaluated over the powers m^m (with 0^0 = 1). See the Mathematica program.

T(n, k) = Sum_{j=0..k-1} (-1)^j*(n-j-1)^(n - 1)/(j! * (k-1-j)!) for 0 <= k < n and T(n, n) = 1.

T(n, k) = r(k-1, n-k, n-k) for n,k >= 1 and T(0, 0) = 1, where r(n, k, m) = m*r(n, k-1, m) + r(n-1, k, m+1) and r(n, 0, m) = 1. (see Vladimir Kruchinin's formula in A039621).

Sum_{k=1..n} binomial(k + x - 1, k-1)*(k-1)!*T(n, k) = (n + x)^(n - 1) for n >= 1.

Sum_{k=1..n} (-1)^(k+j)*Stirling1(k, j)*T(n, k) = n^(n-j)*binomial(n-1, j-1) for n >= 1, which are, up to sign, the coefficients of the Abel polynomials (A137452).

From Werner Schulte, Jun 14 2022 and Jun 19 2022: (Start)

E.g.f. of column k >= 0: (Sum_{i>0} (i-1)^(i-1) * t^i / i!)^k / k!.

Conjecture: T(n, k) = Sum_{i=0..n-k} A048994(n-k, i) * A048993(n+i-1, n-1) for 0 < k <= n and T(n, 0) = 0^n for n >= 0; proved by Mike Earnest, see link at A354797. (End)

A264750 Number of sequences of 5 throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the 5th throw.

Original entry on oeis.org

5, 29, 99, 259, 574, 1134, 2058, 3498, 5643, 8723, 13013, 18837, 26572, 36652, 49572, 65892, 86241, 111321, 141911, 178871, 223146, 275770, 337870, 410670, 495495, 593775, 707049, 836969, 985304, 1153944, 1344904, 1560328, 1802493, 2073813, 2376843, 2714283

Offset: 5

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Author

Louis Rogliano, Nov 23 2015

Keywords

nonn
easy

Comments

Sequence gives the second column of A185508. [Bruno Berselli, Nov 24 2015]

Number of 5-tuples (t_1, ..., t_5) with 1 <= t_j <= n, Sum_{j <= 4} t_j < n and Sum_{j<=5} t_j >= n. - Robert Israel, Nov 25 2015

Examples

From _Jon E. Schoenfield_, Nov 26 2015: (Start) For n=5, the a(5) = 5 sequences (i.e., quintuples or 5-tuples) are {1,1,1,1,1}, {1,1,1,1,2}, {1,1,1,1,3}, {1,1,1,1,4} and {1,1,1,1,5}. (Each of the first four throws must be a 1; otherwise, the sum of the throws would reach or exceed 5 before the 5th throw.) For n=6, each of the quintuples must have four throws whose sum is less than 6, followed by a fifth throw that brings the sum to at least 6, so the a(6) = 29 quintuples are the 5 quintuples {1,1,1,1,t_5} where t_5 is any value in 2..6 and the four sets of 6 quintuples {1,1,1,2,t_5}, {1,1,2,1,t_5}, {1,2,1,1,t_5} and {2,1,1,1,t_5} where t_5 is any value in 1..6. (End)

Links

Colin Barker, Table of n, a(n) for n = 5..1000

Louis Rogliano, Sequence A264750

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000096 (k=2), A051925 (k=3), A215862 (k=4).

Cf. A185508.

Programs

Magma

[(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120: n in [5..40]]; // Vincenzo Librandi, Nov 24 2015

Maple

A264750:=n->(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120: seq(A264750(n), n=5..50); # Wesley Ivan Hurt, Nov 24 2015

Mathematica

f[n_, k_] := Module[ {i, total = 0, part, perm}, part = IntegerPartitions[n, {k}]; perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}], 1]; For[i = 1, i <= Length[perm], i++, total += n + 1 - perm[[i, k]] ]; Return[total]; ] And the sequences are obtained by: h[k_] := Table[f[i, k], {i, k, number_of_terms_wanted}] Table[(n - 4) (n - 3) (n - 2) (n - 1) (4 n + 5)/120, {n, 5, 40}] (* Bruno Berselli, Nov 24 2015 *)

PARI

Vec(x^5*(5-x)/(1-x)^6 + O(x^100)) \\ Colin Barker, Nov 23 2015

PARI

for(n=5, 40, print1((n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120", ")); \\ Bruno Berselli, Nov 24 2015

Sage

[(n-4)*(n-3)*(n-2)*(n-1)*(4*n+5)/120 for n in (5..40)] # Bruno Berselli, Nov 24 2015

Formula

From Colin Barker, Nov 23 2015: (Start)

a(n) = (n - 4)*(n - 3)*(n - 2)*(n - 1)*(4*n + 5)/120.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>10.

G.f.: x^5*(5 - x) / (1 - x)^6. (End)

Extensions

Offset changed by Robert Israel, Nov 25 2015

Formulae, b-file adapted to the new offset and definition rephrased by the Editors of the OEIS, Nov 26 2015

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 19, 9, 1, 0, 16, 65, 55, 14, 1, 0, 32, 211, 285, 125, 20, 1, 0, 64, 665, 1351, 910, 245, 27, 1, 0, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 0, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1

Offset: 0

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Author

Peter Luschny, Apr 10 2016

Keywords

nonn
tabl

Examples

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 19, 9, 1, 0, 16, 65, 55, 14, 1, 0, 32, 211, 285, 125, 20, 1, 0, 64, 665, 1351, 910, 245, 27, 1.

Links

Peter Luschny, Extensions of the binomial

Crossrefs

Variant: A143494 (the main entry for this triangle).

A005493 (row sums), A074051 (alt. row sums), A000079 (col. 1), A001047 (col. 2),

A016269 (col. 3), A025211 (col. 4), A000096 (diag. n,n-1), A215862 (diag. n,n-2),

A049444, A136124, A143491 (matrix inverse).

Cf. A048993, A269951.

Programs

Maple

A269952 := (n,k) -> Stirling2(n+1, k+1) - Stirling2(n, k+1): seq(seq(A269952(n,k), k=0..n), n=0..9);

Mathematica

Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] StirlingS2[j,k], {j,0,n}], {n,0,9}, {k,0,n}]]

Formula

T(n, k) = S2(n+1, k+1) - S2(n, k+1).

A125101 T(n,k) = k*binomial(n-1,k-1) + Fibonacci(k)*binomial(n-1,k) (1 <= k <= n).

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 11, 4, 5, 14, 26, 19, 5, 6, 20, 50, 55, 30, 6, 7, 27, 85, 125, 105, 44, 7, 8, 35, 133, 245, 280, 182, 62, 8, 9, 44, 196, 434, 630, 560, 300, 85, 9, 10, 54, 276, 714, 1260, 1428, 1056, 477, 115, 10, 11, 65, 375, 1110, 2310, 3192, 3030, 1905, 745, 155

Offset: 1

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Author

Gary W. Adamson, Nov 20 2006

Keywords

nonn
tabl

Comments

Row sums are s(n) = 1, 4, 11, 28, 69, 167, 400, ...

Binomial transform of the bidiagonal matrix with (1,2,3...) in the main diagonal and the Fibonacci numbers (1,1,2,3,5,8,...) in the subdiagonal.

Examples

First few rows of the triangle: 1; 2, 2; 3, 5, 3; 4, 9, 11, 4; 5, 14, 26, 19, 5; 6, 20, 50, 55, 30, 6; 7, 27, 85, 125, 105, 44, 7; 8, 35, 133, 245, 280, 182, 62, 8; ...

Crossrefs

Cf. A000096, A051925.

Programs

Maple

with(combinat): T:=(n,k)->k*binomial(n-1,k-1)+fibonacci(k)*binomial(n-1,k): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Mathematica

Flatten[Table[k Binomial[n-1,k-1]+Fibonacci[k]Binomial[n-1,k],{n,15},{k,n}]] (* Harvey P. Dale, Nov 03 2014 *)

Formula

T(n,2) = A000096(n-1).

T(n,3) = A051925(n-1).

T(n,4) = A215862(n-3). - R. J. Mathar, Aug 10 2013

Row sums s(n) = 7*s(n-1) -17*s(n-2) +16*s(n-3) -4*s(n-4) with s(n) = A001787(n+1)/4 +A001906(n-1). - R. J. Mathar, Aug 10 2013

Extensions

Edited by N. J. A. Sloane, Nov 29 2006

A217756 Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes with exactly k components where each component has at most one cycle; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 31, 19, 6, 1, 347, 195, 55, 10, 1, 4956, 2707, 720, 125, 15, 1, 85102, 46319, 12082, 2030, 245, 21, 1, 1698712, 930947, 242774, 40397, 4830, 434, 28, 1, 38562309, 21372678, 5620177, 938826, 112287, 10206, 714, 36, 1

Offset: 1

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Author

Geoffrey Critzer, Mar 23 2013

Keywords

nonn
tabl

Comments

The Bell transform of A129271(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

From Washington Bomfim, May 10 2020: (Start)

The second formula is based on Kolchin's formula (1.4.2) [see the Kolchin reference].

Some special cases of T(n,k) are

Column 2 = n! * Sum_{j=1..floor(n/2)} f(j)/j! * f(n-j)/(n-j)!, odd n.

n!/2 *( (f(n/2)/(n/2)!)^2 + 2 * Sum_{j=1..floor(n/2)-1} f(j)/j! * f(n-j)/(n-j)!), even n.

Diagonal T(n,n-3) = 1/48*n^6 +1/48*n^5 -13/48*n^4 -37/48*n^3 +13/4*n^2 -9/4*n,

Diagonal T(n,n-2) = 1/8*n^4 -1/12*n^3 -5/8*n^2 +7/12*n = A215862(n-2),

Diagonal T(n,n-1) = 1/2*n^2- 1/2*n = A000217(n-1),

and Diagonal T(n,n) = 1. (End)

Examples

... o-o ........... o o ........... o o .......... ... ........... | ........... |\ .......... ... o-o ........... o-o ........... o-o .......... T(4,2) = 19 because the above graphs on 4 nodes have 2 components with at most one cycle. They have respectively 3 + 12 + 4 = 19 labelings. 1; 1, 1; 4, 3, 1; 31, 19, 6, 1; 347, 195, 55, 10, 1; 4956, 2707, 720, 125, 15, 1; 85102, 46319, 12082, 2030, 245, 21, 1;

References

V. F. Kolchin, Random Graphs. Encyclopedia of Mathematics and Its Applications 53. Cambridge Univ. Press, Cambridge, 1999, pp 30-31.

Crossrefs

Row sums = A133686.

Column 1 = A129271.

Cf. A144228, A000217, A215862.

Programs

Mathematica

nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[Exp[y(t/2-3t^2/4)]/(1-t)^(y/2),{x,0,nn}],{x,y}],1]]//Grid

PARI

\p 1000 \\ See Peter Luschny formula in A129271. f(p) = round(((p-1) * exp(p) * incgam(p-1,p) + p^(p-2) * (3-p)) /2); T(n,k) = { my(S=0, D, p, c); forpart(a = n, D = Set(a); S += prod(j=1,#D, p=D[j]; c=#select(x-> x==p,Vec(a)); (f(p)/p!)^c /c!) , [1, n], [k, k]); n! * S }; \\ Washington Bomfim, Jun 16 2020

Sage

# uses[bell_matrix from A264428] # Adds a column 1,0,0,0, ... at the left side of the triangle. bell_matrix(lambda n: A129271(n+1), 10) # Peter Luschny, Jan 18 2016

Formula

E.g.f.: exp(y*A(x)) where A(x) is the e.g.f. for A133686.

T(n,k) = n!/k! * Sum_{compositions p_1 + ... + p_k = n, p_i >= 1} Product_{j=1..k} f(p_j)/p_j!, where f(p)=A129271(p) = ((p-1)*e^p*GAMMA(p-1,p)+p^(p-2)*(3-p))/2.

A264751 Triangle read by rows: T(n,k) is the number of sequences of k <= n throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the k-th throw.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 9, 11, 4, 1, 14, 26, 19, 5, 1, 20, 50, 55, 29, 6, 1, 27, 85, 125, 99, 41, 7, 1, 35, 133, 245, 259, 161, 55, 8, 1, 44, 196, 434, 574, 476, 244, 71, 9, 1, 54, 276, 714, 1134, 1176, 804, 351, 89, 10, 1, 65, 375, 1110, 2058, 2562, 2190, 1275, 485, 109, 11

Offset: 1

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Author

Louis Rogliano, Nov 26 2015

Keywords

nonn
tabl

Comments

By empirical observation: Sum of rows is A002064.

Examples

Triangle begins: 1 1 2 1 5 3 1 9 11 4 1 14 26 19 5 1 20 50 55 29 6 1 27 85 125 99 41 7 1 35 133 245 259 161 55 8 1 44 196 434 574 476 244 71 9 1 54 276 714 1134 1176 804 351 89 10 1 65 375 1110 2058 2562 2190 1275 485 109 11

Links

Cyann Donnot, Antoine Genitrini, Yassine Herida, Unranking Combinations Lexicographically: an efficient new strategy compared with others, hal-02462764 [cs] / [cs.DS] / [math] / [math.CO], 2020.

Antoine Genitrini and Martin Pépin, Lexicographic unranking of combinations revisited, hal-03040740v2 [cs.DM] [cs.DS] [math.CO], 2020.

Crossrefs

Columns are: A000012 (k=1), A000096 (k=2), A051925 (k=3), A215862 (k=4), A264750 (k=5).

Cf. A007318 (binomial(n-1,k-1) = number of sequences of k throws of an n-sided die in which the sum of the throws equals n).

See also A002064.

Programs

Mathematica

T[n_, k_] := Module[ {i, total = 0, part, perm}, part = IntegerPartitions[n, {k}]; perm = Flatten[Table[Permutations[part[[i]]], {i, 1, Length[part]}], 1]; For[i = 1, i <= Length[perm], i++, total += n + 1 - perm[[i, k]] ]; Return[total]; ] (* The rows are obtained by: *) g[n_] := Table[T[n,k], {k,1,n}] (* And the triangle is obtained by: *) Table[g[n],{n,1,number_of_rows_wanted}]

Formula

Sum_{k = 1..n} T(n,k)*k/n^k = ((n+1)/n)^(n-1) = expected value of k.

Lim_{n->infinity} (expected value of k) = e = 2.71828182845... - Jon E. Schoenfield, Nov 26 2015

T(n,k) = Sum_{i=k..n} i*binomial(i-2,k-2). - Danny Rorabaugh, Mar 04 2016

T(n,n-1) = 2*T(n-1,n-1) + T(n-1,n-2).

By empirical observation, g.f. for column k: (x-k)/(x-1)^(k+1).

Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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cp's OEIS Frontend

A215861 Number T(n,k) of simple labeled graphs on n nodes with exactly k connected components that are trees or cycles; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A241765 a(n) = n*(n + 1)*(n + 2)*(3*n + 17)/24.

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Maxima

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A354794 Triangle read by rows. The Bell transform of the sequence {m^m | m >= 0}.

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A264750 Number of sequences of 5 throws of an n-sided die (with faces numbered 1, 2, ..., n) in which the sum of the throws first reaches or exceeds n on the 5th throw.

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A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

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A125101 T(n,k) = k*binomial(n-1,k-1) + Fibonacci(k)*binomial(n-1,k) (1 <= k <= n).

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A241765 a(n) = n(n + 1)(n + 2)(3n + 17)/24.

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

A125101 T(n,k) = kbinomial(n-1,k-1) + Fibonacci(k)binomial(n-1,k) (1 <= k <= n).