A291709 -id:A291709 - cp's OEIS frontend

A293991 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..k+1} binomial(k,j-1)*x^j/j).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 9, 10, 1, 1, 1, 5, 16, 33, 26, 1, 1, 1, 6, 25, 76, 141, 76, 1, 1, 1, 7, 36, 145, 436, 651, 232, 1, 1, 1, 8, 49, 246, 1025, 2776, 3333, 764, 1, 1, 1, 9, 64, 385, 2046, 8245, 19384, 18369, 2620, 1, 1, 1, 10, 81

Offset: 0

Seiichi Manyama, Oct 21 2017

			Square array A(n,k) begins:
   1,  1,   1,   1,    1, ...
   1,  1,   1,   1,    1, ...
   1,  2,   3,   4,    5, ...
   1,  4,   9,  16,   25, ...
   1, 10,  33,  76,  145, ...
   1, 26, 141, 436, 1025, ...

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

Columns k=0..5 give A000012, A000085, A049425, A049426, A049427, A049428.
Rows n=0-1 give A000012.
Main diagonal gives A294003.
Cf. A291709.

A[0, ] = 1; A[n, k_] := (n-1)!*Sum[Binomial[k, j-1]*A[n-j, k]/(n-j)!, {j, 1, Min[k+1, n]}];
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

E.g.f. of column k: exp(((1+x)^(k+1) - 1)/(k+1)).

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

A049376 Row sums of triangle A046089.

Original entry on oeis.org

1, 1, 4, 22, 154, 1306, 12976, 147484, 1883932, 26680924, 414468496, 7001104936, 127677078904, 2498712779512, 52209534323584, 1159559538626896, 27269218041047056, 676732851527182864, 17669429275516846912, 484087943980439097184, 13882791112964223876256

Offset: 0

Wolfdieter Lang

nonn

a(n) is the number of n-permutations where each cycle has two (not necessarily distinct) roots. Here a root means a designated element in a cycle. Cf. A000262 which gives the number of n-permutations with a single root in each cycle. Note that the order of designating the elements is not important. Cf. (A bijection from endofunctions to "doubly" rooted trees where the order of designating the roots is important) Miklos Bona, A Walk Through Combinatorics, World Scientific Publishing, 2006, page 216. - Geoffrey Critzer, May 17 2012.

			a(2) = 4 because we have: (1'')(2'');(1''2);(12'');(1'2') where the permutations are given in cycle notation and the two roots in each cycle are designated by a '.

Alois P. Heinz, Table of n, a(n) for n = 0..436
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=3 of A291709.
Cf. A000085, A000262, A046089, A136658.
Cf. A004211, A375173.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+1)!/2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017
a := proc(n) option remember; `if`(n < 3, [1, 1, 4][n + 1],
a(n-1)*(3*n-2) - a(n-2)*3*(n-1)*(n-2) + a(n-3)*(n-1)*(n-2)*(n-3)) end:
seq(a(n), n=0..20); # after Emanuele Munarini, Peter Luschny, Sep 09 2017

nn = 15;Drop[Range[0, nn]! CoefficientList[Series[Exp[x/(1 - x) + x^2/2/(1 - x)^2], {x, 0, nn}], x], 1]  (* Geoffrey Critzer, May 17 2012 *)

E.g.f.: exp(p(x)) with p(x) := x*(2-x)/(2*(1-x)^2) (E.g.f. first column of A046089).
Lah transform of A000085: a(n) = Sum_{k=0..n} n!/k!*binomial(n-1,k-1) * A000085(k). - Vladeta Jovovic, Oct 02 2003
a(n+3) - (3*n+7)*a(n+2) + 3*(n+1)*(n+2)*a(n+1) - n*(n+1)*(n+2)* a(n) = 0. - Emanuele Munarini, Sep 08 2017
a(n) ~ n^(n-1/6) / sqrt(3) * exp(-1/3 + n^(1/3)/2 + 3*n^(2/3)/2 - n). - Vaclav Kotesovec, Oct 23 2017
E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^3 dx )^n / n!, where the constant of integration is taken to be zero. - Paul D. Hanna, Apr 27 2019
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004211(k).
a(n) = (1/exp(1/2)) * (-1)^n * n! * Sum_{k>=0} binomial(-2*k,n)/(2^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A049377 Row sums of triangle A049352.

Original entry on oeis.org

1, 1, 5, 33, 273, 2721, 31701, 421905, 6302913, 104270913, 1889862021, 37204038081, 789866524305, 17977594555233, 436435929785493, 11251798888929201, 306889765901872641, 8825681949708120705, 266828094135981378693, 8458295877281844310113

Offset: 0

Wolfdieter Lang

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..428
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=4 of A291709.

Cf. A049352, A049425.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+2)!/6*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*(j+2)!/6*a[n-j], {j, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

E.g.f. exp(p(x)) with p(x) := x*(3-3*x+x^2)/(3*(1-x)^3) (E.g.f. first column of A049352).
a(n) ~ n^(n-1/8)/2 * exp(-1/4 + 5*n^(1/4)/24 + sqrt(n)/2 + 4*n^(3/4)/3 - n). - Vaclav Kotesovec, Oct 23 2017
E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^4 dx )^n / n!, where the constant of integration is taken to be zero. - Paul D. Hanna, Apr 27 2019
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004212(k).
a(n) = (1/exp(1/3)) * (-1)^n * n! * Sum_{k>=0} binomial(-3*k,n)/(3^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A049378 Row sums of triangle A049353.

Original entry on oeis.org

1, 1, 6, 46, 436, 4956, 65776, 996976, 16957536, 319259296, 6581662336, 147290942976, 3552885191296, 91827536814976, 2530228890080256, 74003737259670016, 2288810287491774976, 74607500831801289216, 2555587654482227055616, 91746983502042106018816

Offset: 0

Wolfdieter Lang

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..421
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=5 of A291709.

Cf. A049353, A049426.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+3)!/4!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, j - 1]*(j + 3)!/4!*a[n - j], {j, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

E.g.f. exp(p(x)) with p(x) := x*(2-x)*(2-2*x+x^2)/(4*(1-x)^4) (E.g.f. first column of A049353).
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004213(k).
a(n) = (1/exp(1/4)) * (-1)^n * n! * Sum_{k>=0} binomial(-4*k,n)/(4^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A049402 Row sums of triangle A049374.

Original entry on oeis.org

1, 1, 7, 61, 649, 8245, 122215, 2069425, 39328465, 827226505, 19047582055, 475956135205, 12815133759385, 369605936607805, 11361372997850695, 370609338222772825, 12780705695068446625, 464412124831585889425, 17729002673226394402375, 709180766131239680070925

Offset: 0

Wolfdieter Lang

Alois P. Heinz, Table of n, a(n) for n = 0..415
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=6 of A291709.

Cf. A049374, A049427.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+4)!/5!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, j - 1]*(j + 4)!/5!*a[n - j], {j, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

E.g.f. exp(p(x)) with p(x) := (1-(1-x)^5)/(5*(1-x)^5) (E.g.f. first column of A049374).
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A005011(k).
a(n) = (1/exp(1/5)) * (-1)^n * n! * Sum_{k>=0} binomial(-5*k,n)/(5^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A294046 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(1/(1-x)^k - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 13, 0, 1, 4, 21, 68, 73, 0, 1, 5, 36, 195, 580, 501, 0, 1, 6, 55, 424, 2241, 5912, 4051, 0, 1, 7, 78, 785, 6136, 30483, 69784, 37633, 0, 1, 8, 105, 1308, 13705, 104544, 476469, 933200, 394353, 0, 1, 9, 136, 2023, 26748

Offset: 0

Seiichi Manyama, Oct 22 2017

			Square array A(n,k) begins:
   1,   1,    1,     1,      1, ...
   0,   1,    2,     3,      4, ...
   0,   3,   10,    21,     36, ...
   0,  13,   68,   195,    424, ...
   0,  73,  580,  2241,   6136, ...
   0, 501, 5912, 30483, 104544, ...

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

Columns k=0..5 give A000007, A000262, A136658, A202826, A294050, A294051.
Rows n=0..2 give A000012, A001477, A014105.
Main diagonal gives A294047.
Cf. A291709.

A[0, ] = 1; A[n, k_] := k*(n-1)!*Sum[Binomial[j+k-1, 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 *)

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

A294188 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/(1-x)^k - 1)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 9, 28, 13, 0, 1, 16, 117, 256, 73, 0, 1, 25, 336, 1881, 2848, 501, 0, 1, 36, 775, 8416, 35505, 37024, 4051, 0, 1, 49, 1548, 27925, 241696, 763209, 547936, 37633, 0, 1, 64, 2793, 75888, 1134025, 7769856, 18309861, 9064192

Offset: 0

Seiichi Manyama, Oct 24 2017

			Square array A(n,k) begins:
   1,   1,     1,      1,       1, ...
   0,   1,     4,      9,      16, ...
   0,   3,    28,    117,     336, ...
   0,  13,   256,   1881,    8416, ...
   0,  73,  2848,  35505,  241696, ...
   0, 501, 37024, 763209, 7769856, ...

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

Columns k=0..3 give A000007, A000262, A294189, A294190.
Rows n=0..1 give A000012, A000290.
Main diagonal gives A294192.
Cf. A291709, A294046.

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

A293986 E.g.f.: exp((1/(1-x)^7 - 1)/7).

Original entry on oeis.org

1, 1, 9, 97, 1249, 19041, 338281, 6872929, 156878337, 3964344769, 109649487241, 3290138609121, 106344130171489, 3680908100971297, 135759417307847529, 5312294503297893601, 219710817062641086721, 9572694339517999983489, 438078249108592685152777

Offset: 0

Seiichi Manyama, Oct 21 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..404

Column k=8 of A291709.

N=66; x='x+O('x^N); Vec(serlaplace(exp((1/(1-x)^7-1)/7)))

A293987 E.g.f.: exp((1/(1-x)^8 - 1)/8).

Original entry on oeis.org

1, 1, 10, 118, 1648, 27136, 520336, 11417680, 281653408, 7694191648, 230059907776, 7461932378176, 260684280697600, 9752025371845888, 388715094054028288, 16438054175878813696, 734700126836545463296, 34591098659155269944320, 1710537337734107030161408

Offset: 0

Seiichi Manyama, Oct 21 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..400

Column k=9 of A291709.

N=66; x='x+O('x^N); Vec(serlaplace(exp((1/(1-x)^8-1)/8)))

A293988 E.g.f.: exp((1/(1-x)^9 - 1)/9).

Original entry on oeis.org

1, 1, 11, 141, 2121, 37461, 769851, 18116001, 479593521, 14065160841, 451507161771, 15721216032501, 589564030968441, 23674308696244701, 1012920637676448411, 45978351344438052681, 2205805135900188831201, 111472339036480856136081

Offset: 0

Seiichi Manyama, Oct 21 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..396

Column k=10 of A291709.

With[{nn=20},CoefficientList[Series[Exp[(1/(1-x)^9-1)/9],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 19 2019 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp((1/(1-x)^9-1)/9)))

cp's OEIS Frontend

A293991 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..k+1} binomial(k,j-1)*x^j/j).

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A049376 Row sums of triangle A046089.

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A049377 Row sums of triangle A049352.

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A049378 Row sums of triangle A049353.

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A049402 Row sums of triangle A049374.

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A294046 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(1/(1-x)^k - 1).

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A294188 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/(1-x)^k - 1)).

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A293986 E.g.f.: exp((1/(1-x)^7 - 1)/7).

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