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Search: a309386 -id:a309386
Displaying 1-5 of 5 results found.
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A317996
Expansion of e.g.f. exp((1 - exp(-3*x))/3).
+10
13
1, 1, -2, 1, 19, -128, 379, 1549, -32600, 261631, -845909, -10713602, 237695149, -2513395259, 11792378662, 151915180429, -4826456213273, 70741388773960, -558513179369297, -2833805536521839, 200720356696607416, -4256279445015662093, 54120395442382043743, -173423789950999240226
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..495
Eric Weisstein's World of Mathematics, Bell Polynomial
FORMULA
a(n) = Sum_{k=0..n} (-3)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-3)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-3)^n BellPolynomial_n(-1/3). - Peter Luschny, Aug 20 2018
MAPLE
a:=series(exp((1 - exp(-3*x))/3), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # Paolo P. Lava, Mar 26 2019
MATHEMATICA
nmax = 23; CoefficientList[Series[Exp[(1 - Exp[-3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-3)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = Sum[(-3)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
Table[(-3)^n BellB[n, -1/3], {n, 0, 23}] (* Peter Luschny, Aug 20 2018 *)
CROSSREFS
Column k=3 of A309386.
Cf. A004212, A007559, A009235, A014182, A318179, A318180, A318181.
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Aug 20 2018
STATUS
approved
A318179
Expansion of e.g.f. exp((1 - exp(-4*x))/4).
+10
10
1, 1, -3, 5, 25, -343, 2133, -3603, -112975, 1938897, -18008275, 55198805, 1753746377, -45801271943, 649021707397, -4682002329795, -50792700319903, 2692784088681889, -59182401177647011, 801759226622986917, -2169423359710146183, -263145142263538606519, 9869607872225170545333
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..474
Eric Weisstein's World of Mathematics, Bell Polynomial
FORMULA
a(n) = Sum_{k=0..n} (-4)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-4)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-4)^n*BellPolynomial_n(-1/4). - Peter Luschny, Aug 20 2018
MAPLE
seq(n!*coeff(series(exp((1-exp(-4*x))/4), x=0, 23), x, n), n=0..22); # Paolo P. Lava, Jan 09 2019
MATHEMATICA
nmax = 22; CoefficientList[Series[Exp[(1 - Exp[-4 x])/4], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-4)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = Sum[(-4)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
Table[(-4)^n BellB[n, -1/4], {n, 0, 22}] (* Peter Luschny, Aug 20 2018 *)
CROSSREFS
Column k=4 of A309386.
Cf. A004213, A007696, A009235, A014182, A317996, A318180, A318181.
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Aug 20 2018
STATUS
approved
A292861
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 - exp(x))).
+10
9
1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, 1, 0, 1, -4, 6, 2, 1, 0, 1, -5, 12, -3, -6, -2, 0, 1, -6, 20, -20, -21, -14, -9, 0, 1, -7, 30, -55, -20, 24, 26, -9, 0, 1, -8, 42, -114, 45, 172, 195, 178, 50, 0, 1, -9, 56, -203, 246, 370, 108, -111, 90, 267, 0, 1, -10, 72, -328, 679, 318, -1105, -2388, -3072, -2382, 413, 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 * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} (-k)^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
A(n,k) = BellPolynomial(n, -k). - Peter Luschny, Dec 23 2021
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, -1, -2, -3, -4, -5, -6, ...
0, 0, 2, 6, 12, 20, 30, ...
0, 1, 2, -3, -20, -55, -114, ...
0, 1, -6, -21, -20, 45, 246, ...
0, -2, -14, 24, 172, 370, 318, ...
0, -9, 26, 195, 108, -1105, -4074, ...
MAPLE
A:= proc(n, k) option remember; `if`(n=0, 1,
-(1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
end:
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Sep 25 2017
MATHEMATICA
A[n_, k_] := Sum[(-k)^j StirlingS2[n, j], {j, 0, n}];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 10 2021 *)
A292861[n_, k_] := BellB[k, k - n];
Table[A292861[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)
CROSSREFS
Columns k=0..4 give A000007, A000587, A213170, A309084, A309085.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292866.
Cf. A292860, A309386.
KEYWORD
sign,tabl,look
AUTHOR
Seiichi Manyama, Sep 25 2017
STATUS
approved
A306245
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} k^j * binomial(n-1,j) * A(j,k) for n > 0.
+10
5
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 17, 15, 1, 1, 1, 5, 43, 179, 52, 1, 1, 1, 6, 89, 1279, 3489, 203, 1, 1, 1, 7, 161, 5949, 108472, 127459, 877, 1, 1, 1, 8, 265, 20591, 1546225, 26888677, 8873137, 4140, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
LINKS
Seiichi Manyama, Antidiagonals n = 0..55, flattened
FORMULA
G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(k * x / (1 - x)) / (1 - x). - Seiichi Manyama, Jun 18 2022
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
1, 5, 17, 43, 89, 161, ...
1, 15, 179, 1279, 5949, 20591, ...
1, 52, 3489, 108472, 1546225, 12950796, ...
MAPLE
A:= proc(n, k) option remember; `if`(n=0, 1,
add(k^j*binomial(n-1, j)*A(j, k), j=0..n-1))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Jul 28 2019
MATHEMATICA
A[0, _] = 1;
A[n_, k_] := A[n, k] = Sum[k^j Binomial[n-1, j] A[j, k], {j, 0, n-1}];
Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 29 2020 *)
CROSSREFS
Columns k=0..4 give A000012, A000110, A126443, A355081, A355082.
Rows n=0+1, 2 give A000012, A000027(n+1).
Main diagonal gives A309401.
Cf. A309386.
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jul 28 2019
STATUS
approved
A334192
Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(1/k) * Sum_{j>=0} (k*j + 1)^n / ((-k)^j * j!).
+10
3
1, 1, 0, 1, 0, -1, 1, 0, -2, -1, 1, 0, -3, -4, 2, 1, 0, -4, -9, 4, 9, 1, 0, -5, -16, 0, 64, 9, 1, 0, -6, -25, -16, 189, 248, -50, 1, 0, -7, -36, -50, 384, 1377, 48, -267, 1, 0, -8, -49, -108, 625, 4416, 4374, -6512, -413, 1, 0, -9, -64, -196, 864, 10625, 26368, -26001, -51200, 2180
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
LINKS
Table of n, a(n) for n=0..65.
FORMULA
G.f. of column k: (1/(1 - x)) * Sum_{j>=0} (-x/(1 - x))^j / Product_{i=1..j} (1 - k*i*x/(1 - x)).
E.g.f. of column k: exp(x + (1 - exp(k*x)) / k).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 0, 0, 0, 0, 0, ...
-1, -2, -3, -4, -5, -6, ...
-1, -4, -9, -16, -25, -36, ...
2, 4, 0, -16, -50, -108, ...
9, 64, 189, 384, 625, 864, ...
MATHEMATICA
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(-x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[x + (1 - Exp[k x])/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
CROSSREFS
Columns k=1..3 give A293037, A334190, A334191.
Cf. A309386, A334165, A334193 (diagonal).
KEYWORD
sign,tabl
AUTHOR
Ilya Gutkovskiy, Apr 18 2020
STATUS
approved
page 1

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