A321086
O.g.f. A(x) satisfies: [x^n] exp(n*A(x)) * (1 - n*x - n*x^2) = 0, for n > 0.
Original entry on oeis.org
1, 2, 6, 32, 220, 1812, 17108, 180512, 2093760, 26396160, 358741328, 5223336288, 81079811280, 1336407320080, 23311138957200, 429063111959808, 8311760620707648, 169072470759431232, 3603666131945918144, 80327823251439861760, 1869212211081119135616, 45331401566332423284864, 1143967734536203174726784, 29996686272924492809481216, 816185909551276017516640000
Offset: 1
O.g.f.: A(x) = x + 2*x^2 + 6*x^3 + 32*x^4 + 220*x^5 + 1812*x^6 + 17108*x^7 + 180512*x^8 + 2093760*x^9 + 26396160*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) * (1 - n*x - n*x^2) begins:
n=1: [1, 0, 1, 28, 729, 26416, 1321225, 87466716, ...];
n=2: [1, 0, 0, 32, 1200, 49152, 2569600, 172974720, ...];
n=3: [1, 0, -3, 0, 1089, 60408, 3509325, 246760776, ...];
n=4: [1, 0, -8, -80, 0, 49024, 3777280, 293683968, ...];
n=5: [1, 0, -15, -220, -2535, 0, 2848825, 291386100, ...];
n=6: [1, 0, -24, -432, -7056, -105984, 0, 208089216, ...];
n=7: [1, 0, -35, -728, -14175, -293048, -5733875, 0, ...];
n=8: [1, 0, -48, -1120, -24576, -590592, -15603200, -391709184, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
(a) Differential Equation.
O.g.f. A(x) satisfies: A(x) = x + x^2 + x*A(x)*A'(x) where
A'(x) = 1 + 4*x + 18*x^2 + 128*x^3 + 1100*x^4 + 10872*x^5 + 119756*x^6 + ...
A(x)*A'(x) = x + 6*x^2 + 32*x^3 + 220*x^4 + 1812*x^5 + 17108*x^6 + 17108*x^7 + ...
so that A(x) - x*A(x)*A'(x) = x + x^2.
(b) Exponentiation.
exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 985*x^4/4! + 32321*x^5/5! + 1544701*x^6/6! + 99637105*x^7/7! + 8257877489*x^8/8! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 25*x^3/3! - 599*x^4/4! - 21681*x^5/5! - 1106939*x^6/6! - 74873737*x^7/7! - 6431021295*x^8/8! + ...
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{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = -Vec( exp(m*x*Ser(A))*(1-m*x-m*x^2 +x^2*O(x^m)))[m+1]/m ); A[n]}
for(n=1, 30, print1(a(n), ", "))
A385766
G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x*A(x)^2 - x^2*A(x)*A'(x))).
Original entry on oeis.org
1, 2, 9, 66, 629, 7071, 89609, 1248355, 18820831, 303879698, 5215803877, 94656100969, 1808853399445, 36282216181916, 761902799960049, 16714472406574829, 382369378451581045, 9107117241193913850, 225512045313741357841, 5798133159909683869788
Offset: 0
-
terms = 20; A[] = 0; Do[A[x] = 1/((1-x)*(1-x*A[x]^2-x^2*A[x]*A'[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
A385767
G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x*A(x)^3 - x^2*A(x)^2*A'(x))).
Original entry on oeis.org
1, 2, 11, 103, 1240, 17405, 272647, 4652676, 85204285, 1657791964, 34030090459, 733238701637, 16520229963511, 388058679087053, 9481616930642904, 240524381652918706, 6324953229391777117, 172191111285984106951, 4847629590517906310392, 140987258808372483601766
Offset: 0
-
terms = 20; A[] = 0; Do[A[x] = 1/((1-x)*(1-x*A[x]^3-x^2*A[x]^2*A'[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
A386229
G.f. A(x) satisfies A(x) = 1/( (1-x)^2 * (1 - x*A(x) - x^2*A'(x)) ).
Original entry on oeis.org
1, 3, 12, 70, 535, 4908, 51478, 600584, 7662285, 105684465, 1563183259, 24645719004, 412279514088, 7290426692472, 135862518564330, 2661378323466016, 54675576786754501, 1175673956931922257, 26411686616265112230, 618863341216409971750, 15101129008183181824938
Offset: 0
-
terms = 21; A[] = 1; Do[A[x] = 1/((1-x)^2(1-x*A[x]-x^2*A'[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 16 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(i+1)*(1+sum(j=0, i-1, v[j+1]*v[i-j])/2)); v;
A386230
G.f. A(x) satisfies A(x) = 1/( (1-x)^3 * (1 - x*A(x) - x^2*A'(x)) ).
Original entry on oeis.org
1, 4, 18, 114, 945, 9399, 106645, 1342028, 18409725, 272154510, 4300884555, 72225827628, 1283066570500, 24025524690426, 472822444534395, 9755834028122904, 210600429263424372, 4747647482075588598, 111583282733838959542, 2729989048854423409090, 69430953497076613542366
Offset: 0
-
terms = 21; A[] = 1; Do[A[x] = 1/((1-x)^3(1-x*A[x]-x^2*A'[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 16 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=binomial(i+2, 2)+(i+1)/2*sum(j=0, i-1, v[j+1]*v[i-j])); v;
A386238
G.f. A(x) satisfies A(x) = 1/( (1+x) * (1 - x*A(x) - x^2*A'(x)) ).
Original entry on oeis.org
1, 0, 1, 3, 16, 98, 708, 5827, 53758, 548879, 6142379, 74756975, 983186673, 13897204929, 210128473576, 3384688592135, 57868599692392, 1046744442623159, 19972732558110246, 400941834927714249, 8447457715944191407, 186387757159393453454, 4298157267300757437962
Offset: 0
-
terms = 23; A[] = 1; Do[A[x] = 1/((1+x)(1-x*A[x]-x^2*A'[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 16 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+(i+1)/2*sum(j=0, i-1, v[j+1]*v[i-j])); v;
A376126
a(n) = 1 + Sum_{k=0..n-1} (k+1)^2 * a(k) * a(n-k-1).
Original entry on oeis.org
1, 2, 11, 127, 2446, 69766, 2758136, 144660949, 9742594594, 820936644105, 84720182796109, 10518406503248799, 1547501158599695053, 266325797207877471893, 53021169588315247959918, 12092601263851930181412631, 3132724961425012148414534036, 914922971820526793968713748093
Offset: 0
-
a[n_] := a[n] = 1 + Sum[(k + 1)^2 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x] - 3 x^2 A'[x] - x^3 A''[x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
A376127
a(n) = 1 + Sum_{k=0..n-1} (k+1)^3 * a(k) * a(n-k-1).
Original entry on oeis.org
1, 2, 19, 565, 38056, 4886164, 1071397370, 370880032881, 191040201050842, 139853547948358801, 140279102716474353325, 187136598610376840549341, 323937672908434382002891895, 712668454800648677607151322833, 1957709831409075714559805601326566, 6613164804688226108094777888275765585
Offset: 0
-
a[n_] := a[n] = 1 + Sum[(k + 1)^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x] - 7 x^2 A'[x] - 6 x^3 A''[x] - x^4 A'''[x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
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