A355113
Expansion of e.g.f. 5 / (6 - 5*x - exp(5*x)).
Original entry on oeis.org
1, 2, 13, 133, 1779, 29565, 589705, 13728695, 365295695, 10934634985, 363678872325, 13305294463275, 531030788556475, 22960273845453725, 1069101897816615425, 53336480697298243375, 2838300249311563302375, 160480124820425410172625, 9607441647405962075600125
Offset: 0
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nmax = 18; CoefficientList[Series[5/(6 - 5 x - Exp[5 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 5^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
A355114
Expansion of e.g.f. 6 / (7 - 6*x - exp(6*x)).
Original entry on oeis.org
1, 2, 14, 156, 2256, 40416, 869040, 21817440, 626063616, 20210176512, 724888631808, 28599923045376, 1230970377166848, 57397448756994048, 2882187551571941376, 155065468075097960448, 8898907099302329647104, 542609247778976191610880, 35031706496702707368591360
Offset: 0
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nmax = 18; CoefficientList[Series[6/(7 - 6 x - Exp[6 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 6^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
A367837
Expansion of e.g.f. 1/(2 - x - exp(4*x)).
Original entry on oeis.org
1, 5, 66, 1294, 33752, 1100504, 43060176, 1965653232, 102548623744, 6018735869824, 392498702352128, 28155539333730560, 2203322337542003712, 186790304541786160128, 17053569926181643921408, 1668166923908523824576512, 174057374767036007615922176
Offset: 0
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 4^j*binomial(i, j)*v[i-j+1])); v;
A352293
Expansion of e.g.f. 1/(2 - exp(x) - x/(1 + x)).
Original entry on oeis.org
1, 2, 7, 43, 335, 3301, 38925, 535851, 8429139, 149173321, 2933274593, 63446532271, 1497102036567, 38269877372637, 1053531222709269, 31074273060116083, 977649690943993979, 32680936703516606737, 1156722832021068313833, 43216064601701505904983
Offset: 0
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m = 19; Range[0, m]! * CoefficientList[Series[1/(2 - Exp[x] - x/(1 + x)), {x, 0, m}], x] (* Amiram Eldar, Mar 11 2022 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-x/(1+x))))
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a(n) = if(n==0, 1, sum(k=1, n, ((-1)^(k-1)*k!+1)*binomial(n, k)*a(n-k)));
A352299
Expansion of e.g.f. 1/(2 - exp(x) - x^3).
Original entry on oeis.org
1, 1, 3, 19, 123, 1021, 10683, 127093, 1725867, 26535613, 452307243, 8475606613, 173390108235, 3842119808749, 91675559886459, 2343875745873493, 63920729617231275, 1852126733351677021, 56823327291638414667, 1840195730889731550805
Offset: 0
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m = 19; Range[0, m]! * CoefficientList[Series[1/(2 - Exp[x] - x^3), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-x^3)))
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b(n, m) = if(n==0, 1, sum(k=1, n, (1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 3);
A352300
Expansion of e.g.f. 1/(2 - exp(x) - x^4).
Original entry on oeis.org
1, 1, 3, 13, 99, 781, 7563, 84253, 1103595, 16074589, 260443083, 4630046653, 90017588235, 1894771249021, 42957132108075, 1043136555486493, 27024421701469995, 743851294350730141, 21679544916491784843, 666932347454809048189
Offset: 0
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m = 19; Range[0, m]! * CoefficientList[Series[1/(2 - Exp[x] - x^4), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-x^4)))
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b(n, m) = if(n==0, 1, sum(k=1, n, (1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 4);
A352306
Expansion of e.g.f. 1/(2 - exp(x) - x^2/2).
Original entry on oeis.org
1, 1, 4, 19, 129, 1071, 10743, 125455, 1675439, 25167073, 420070323, 7712503173, 154475622513, 3351859639363, 78324320723561, 1960968388497523, 52368881358012435, 1485952518531483045, 44643697199669589447, 1415782273405809697009
Offset: 0
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m = 19; Range[0, m]! * CoefficientList[Series[1/(2 - Exp[x] - x^2/2), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-x^2/2)))
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b(n, m) = if(n==0, 1, sum(k=1, n, (1+(k==m))*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 2);
A352307
Expansion of e.g.f. 1/(2 - exp(x) - x^3/6).
Original entry on oeis.org
1, 1, 3, 14, 83, 621, 5583, 58493, 700507, 9438253, 141291843, 2326680313, 41797029035, 813422096709, 17047913249279, 382815685896293, 9169316015977675, 233352842701661021, 6288004372005738747, 178851946015229702545, 5354894260179239755995
Offset: 0
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m = 20; Range[0, m]! * CoefficientList[Series[1/(2 - Exp[x] - x^3/6), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-x^3/6)))
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b(n, m) = if(n==0, 1, sum(k=1, n, (1+(k==m))*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 3);
A367830
E.g.f. A(x) satisfies A(x) = (1 + (exp(x) - 1) * A(2*x)) / (1 - x).
Original entry on oeis.org
1, 2, 13, 208, 7817, 681626, 136872113, 62739300968, 64993463748977, 150619722938940622, 773428868899900772345, 8724654696222415759129388, 214574098061440421518595200025, 11429824974654804201081062775335234, 1311103770238649103823410558613476172193
Offset: 0
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^(i-j)*binomial(i, j)*v[i-j+1])); v;
A367831
E.g.f. A(x) satisfies A(x) = (1 + (exp(x) - 1) * A(3*x)) / (1 - x).
Original entry on oeis.org
1, 2, 17, 529, 60191, 24822701, 36413854321, 186201636968159, 3260017071214457747, 192544750449664642891369, 37901471231124512743264725077, 24619109083914012570141331273785011, 52334858943702505364559907161989713988743
Offset: 0
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 3^(i-j)*binomial(i, j)*v[i-j+1])); v;