A293037 -id:A293037 - cp's OEIS frontend

A196835 Alternating row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

1, 4, 15, 51, 146, 273, -319, -6374, -36235, -113833, 69388, 3772035, 28631669, 112704452, -96418909, -5652669753, -50538496446, -230554460867, 281597003109, 16303457144146, 166512491229617, 872578914956059, -1111135578108284, -78512971676777833, -919653124088665479

Offset: 0

Wolfdieter Lang, Oct 07 2011

			a(2) = 25 - 11 + 1 = 15.

Seiichi Manyama, Table of n, a(n) for n = 0..589

Cf. A193685, A196834 (row sums), A193683, A193684, A293037, A346740.

my(x='x+O('x^30)); Vec(serlaplace(exp(-exp(x)+5*x+1))) \\ Michel Marcus, Aug 02 2021

a(n) = Sum_{m=0..n} (-1)^m * A193685(n,m), n>=0.
E.g.f.: exp(-exp(x)+5*x+1).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 5)^n / k!. - Ilya Gutkovskiy, Dec 20 2019
a(0) = 1; a(n) = 5 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

A193683 Alternating row sums of Sheffer triangle A143495 (3-restricted Stirling2 numbers).

Original entry on oeis.org

1, 2, 3, 1, -14, -59, -99, 288, 2885, 10365, 1700, -226313, -1535203, -4258630, 17243695, 284513877, 1688253890, 2750940953, -51540956455, -624352447488, -3470378651847, -496964048927, 204678286709292, 2311290490508227, 12611758414937801

Offset: 0

Wolfdieter Lang, Oct 06 2011

In order to have a lower triangular Sheffer matrix for A143495 one uses row and column offsets 0 (not 3).

			Row no. 3 of A143495 with [0,0] offset is [27,37,12,1], hence a(3)=27-37+12-1=1.

See A143495.

Seiichi Manyama, Table of n, a(n) for n = 0..591

Cf. A143495, A074051 (2-restricted Stirling2 case), A193684, A196835, A293037, A346738.

With[{nn=30},CoefficientList[Series[Exp[3x+1-Exp[x]],{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Jan 10 2013 *)

E.g.f.: exp(-exp(x)+3*x+1).
G.f.: (1 - 2/E(0))/x where E(k) = 1 + 1/(1 - 2*x/(1 - 2*(k+1)*x/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
G.f.: 1/U(0) where U(k) = 1 - x*(k+2) + x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: (1 - G(0) )/(x+1) where G(k) =  1 - 1/(1-k*x-3*x)/(1-x/(x+1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 17 2013
G.f.: 1/Q(0), where Q(k) = 1 - 3*x + x/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: G(0)/(1-2*x), where G(k) = 1 - x^2*(2*k+1)/(x^2*(2*k+1) + (1-x*(2*k+2))*(1-x*(2*k+3))/(1 - x^2*(2*k+2)/(x^2*(2*k+2) + (1-x*(2*k+3))*(1-x*(2*k+4))/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Feb 06 2014
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 3)^n / k!. - Ilya Gutkovskiy, Dec 20 2019
a(0) = 1; a(n) = 3 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

A193684 Alternating row sums of Sheffer triangle A143496 (4-restricted Stirling2 numbers).

Original entry on oeis.org

1, 3, 8, 17, 17, -78, -585, -2021, -1710, 29395, 231413, 856264, -346979, -30019585, -232782792, -834712259, 2313820717, 59793779314, 469729578123, 1597321309383, -9914171906614, -206169178856073, -1697255630380351, -5677886943413120, 55801423903125353

Offset: 0

Wolfdieter Lang, Oct 06 2011

In order to have A143496 as a lower triangular Sheffer matrix one uses row and column offsets 0 (not 4).

			With offset [0,0] row n=3 of A143496 is [64,61,15,1], hence a(3)=64-61+15-1=17.

Seiichi Manyama, Table of n, a(n) for n = 0..591

Cf. A143496, A193683 (3-restricted Stirling2 case), A196835, A293037, A346739.

my(x='x+O('x^30)); Vec(serlaplace(exp(-exp(x)+4*x+1))) \\ Michel Marcus, Aug 02 2021

E.g.f.: exp(-exp(x)+4*x+1).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 4)^n / k!. - Ilya Gutkovskiy, Dec 20 2019
a(0) = 1; a(n) = 4 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

A293038 E.g.f.: exp(1 + x + x^2/2! - exp(x)).

Original entry on oeis.org

1, 0, 0, -1, -1, -1, 9, 34, 90, -71, -1645, -9439, -25367, 45902, 1070146, 7122361, 24063637, -54352333, -1501032375, -12319959348, -53177369044, 80189626539, 3910291080509, 40317032441401, 228707685648269, 38882013140648, -16392939262378536

Offset: 0

Seiichi Manyama, Sep 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..592

Column k=2 of A293051.
Cf. A000587 (k=0), A293037 (k=1), this sequence (k=2), A293039 (k=3), A293040 (k=4).
Cf. A006505.

seq(factorial(n) * coeftayl(exp(1+x+x^2/2!-exp(x)), x = 0, n),n = 0..50); # Muniru A Asiru, Oct 05 2017

my(x='x+O('x^66)); Vec(serlaplace(exp(-exp(x)+1+x+x^2/2)))

a(0) = 1; a(n) = -Sum_{k=3..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Nov 20 2020

A335977 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(1 - exp(x)) + x).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -1, -1, 1, 1, -3, 1, 3, 2, 1, 1, -4, 5, 7, 7, 9, 1, 1, -5, 11, 5, -8, -13, 9, 1, 1, -6, 19, -9, -43, -65, -89, -50, 1, 1, -7, 29, -41, -74, -27, 37, -45, -267, 1, 1, -8, 41, -97, -53, 221, 597, 1024, 1191, -413, 1, 1, -9, 55, -183, 92, 679, 961, 805, 1351, 4723, 2180, 1

Offset: 0

Seiichi Manyama, Jul 03 2020

			Square array begins:
  1,  1,   1,   1,   1,   1,    1, ...
  1,  0,  -1,  -2,  -3,  -4,   -5, ...
  1, -1,  -1,   1,   5,  11,   19, ...
  1, -1,   3,   7,   5,  -9,  -41, ...
  1,  2,   7,  -8, -43, -74,  -53, ...
  1,  9, -13, -65, -27, 221,  679, ...
  1,  9, -89,  37, 597, 961, -341, ...

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

Columns k=0-4 give: A000012, A293037, A309775, A320432, A320433.
Main diagonal gives A334241.
Cf. A292861, A335975.

T[0, k_] := 1; T[n_, k_] := T[n - 1, k] - k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)

T(0,k) = 1 and T(n,k) = T(n-1,k) - k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

T(n,k) = exp(k) * Sum_{j>=0} (j + 1)^n * (-k)^j / j!.

A293039 E.g.f.: exp(1 + x + x^2/2! + x^3/3! - exp(x)).

Original entry on oeis.org

1, 0, 0, 0, -1, -1, -1, -1, 34, 125, 335, 791, -4027, -41328, -223510, -966174, -1082043, 22493107, 255137121, 1853859145, 8611832136, 6734302429, -364364045001, -4974309134233, -41550393316275, -223452696895652, -173393115915136, 14282249293678744

Offset: 0

Seiichi Manyama, Sep 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..592

Column k=3 of A293051.
Cf. A000587 (k=0), A293037 (k=1), A293038 (k=2), this sequence (k=3), A293040 (k=4).
Cf. A057837.

With[{nn=30},CoefficientList[Series[Exp[1+x+x^2/2!+x^3/3!-Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 19 2022 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(-exp(x)+1+x+x^2/2+x^3/6)))

a(0) = 1; a(n) = -Sum_{k=4..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Nov 20 2020

A293040 E.g.f.: exp(1 + x + x^2/2! + x^3/3! + x^4/4! - exp(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, -1, -1, -1, -1, -1, 125, 461, 1253, 3002, 6720, -111684, -978758, -5246983, -22948029, -89534309, 164027151, 5722510249, 55413784239, 393256686307, 2377996545081, 7807749195198, -46231762188586, -1125536160278906, -12849721017510166

Offset: 0

Seiichi Manyama, Sep 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..593

Column k=4 of A293051.
Cf. A000587 (k=0), A293037 (k=1), A293038 (k=2), A293039 (k=3), this sequence (k=4).
Cf. A057814.

seq(factorial(n)*coeftayl(exp(1+x+x^2/2!+x^3/3!+x^4/4!-exp(x)), x = 0, n),n=0..50); # Muniru A Asiru, Oct 06 2017

my(x='x+O('x^66)); Vec(serlaplace(exp(-exp(x)+1+x+x^2/2+x^3/6+x^4/24)))

a(0) = 1; a(n) = -Sum_{k=5..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Nov 20 2020

A293051 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{i=0..k} x^i/i! - exp(x)).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, 1, 0, -1, 1, 1, 0, 0, -1, 1, 1, 0, 0, -1, 2, -2, 1, 0, 0, 0, -1, 9, -9, 1, 0, 0, 0, -1, -1, 9, -9, 1, 0, 0, 0, 0, -1, 9, -50, 50, 1, 0, 0, 0, 0, -1, -1, 34, -267, 267, 1, 0, 0, 0, 0, 0, -1, -1, 90, -413, 413, 1, 0, 0, 0, 0, 0, -1, -1, 34, -71

Offset: 0

Seiichi Manyama, Sep 29 2017

			Square array begins:
    1,  1,  1,  1,  1, ...
   -1,  0,  0,  0,  0, ...
    0, -1,  0,  0,  0, ...
    1, -1, -1,  0,  0, ...
    1,  2, -1, -1,  0, ...
   -2,  9, -1, -1, -1, ...

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

Columns k=0..4 give A000587, A293037, A293038, A293039, A293040.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
Cf. A229223, A293024.

E.g.f. of column k: Product_{i>k} exp(-x^i/i!).

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = - Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k.

A334190 a(n) = exp(1/2) * Sum_{k>=0} (2k + 1)^n / ((-2)^k k!).

Original entry on oeis.org

1, 0, -2, -4, 4, 64, 248, 48, -6512, -51200, -171296, 830400, 17870400, 144684032, 441316224, -5976726784, -119879356160, -1123892297728, -3962230563328, 70410917051392, 1686366492509184, 19578100126072832, 101728414306826240, -1258662784047370240, -42727186269262737408

Offset: 0

Ilya Gutkovskiy, Apr 18 2020

sign

Column k=2 of A334192.

Cf. A007405, A009235, A293037, A308536, A334191.

nmax = 24; CoefficientList[Series[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - 2 j x/(1 - x)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[x + (1 - Exp[2 x])/2], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k, -1/2], {k, 0, n}], {n, 0, 24}] (* Vaclav Kotesovec, Apr 18 2020 *)

G.f.: (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - 2*j*x/(1 - x)).

E.g.f.: exp(x + (1 - exp(2*x)) / 2).

A334191 a(n) = exp(1/3) * Sum_{k>=0} (3k + 1)^n / ((-3)^k k!).

Original entry on oeis.org

1, 0, -3, -9, 0, 189, 1377, 4374, -26001, -560601, -4999482, -18631053, 235966365, 5966310960, 71037580689, 407585191059, -3965310883512, -157871090202975, -2631946996862451, -24922384546473810, 45577755305571339, 7795795206234609027, 192159735553383097014

Offset: 0

Ilya Gutkovskiy, Apr 18 2020

sign

Column k=3 of A334192.

Cf. A003575, A293037, A317996, A334190.

nmax = 22; CoefficientList[Series[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - 3 j x/(1 - x)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Exp[x + (1 - Exp[3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] * 3^k * BellB[k, -1/3], {k, 0, n}], {n, 0, 22}] (* Vaclav Kotesovec, Apr 18 2020 *)

G.f.: (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - 3*j*x/(1 - x)).

E.g.f.: exp(x + (1 - exp(3*x)) / 3).

cp's OEIS Frontend

A196835 Alternating row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A193683 Alternating row sums of Sheffer triangle A143495 (3-restricted Stirling2 numbers).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A193684 Alternating row sums of Sheffer triangle A143496 (4-restricted Stirling2 numbers).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A293038 E.g.f.: exp(1 + x + x^2/2! - exp(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A335977 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(1 - exp(x)) + x).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A293039 E.g.f.: exp(1 + x + x^2/2! + x^3/3! - exp(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A293040 E.g.f.: exp(1 + x + x^2/2! + x^3/3! + x^4/4! - exp(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A293051 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{i=0..k} x^i/i! - exp(x)).

Views

Author

Keywords

Examples

Links

A334190 a(n) = exp(1/2) * Sum_{k>=0} (2k + 1)^n / ((-2)^k k!).

A334191 a(n) = exp(1/3) * Sum_{k>=0} (3k + 1)^n / ((-3)^k k!).