A339807
Irregular triangle read by rows: T(n,k) (n>=2, k>=1) is the number of strong digraphs on n nodes with k descents.
Original entry on oeis.org
1, 2, 11, 5, 10, 154, 540, 581, 272, 49, 122, 3418, 27304, 90277, 150948, 150519, 95088, 37797, 8714, 893, 3346, 142760, 1938178, 12186976, 42696630, 94605036, 145009210, 161845163, 134933733, 84656743, 39632149, 13481441, 3156845, 455917, 30649
Offset: 2
Triangle begins:
1;
2, 11, 5;
10, 154, 540, 581, 272, 49;
122, 3418, 27304, 90277, 150948, 150519, 95088, 37797, 8714, 893;
3346, 142760, 1938178, 12186976, 42696630, 94605036, 145009210, 161845163, 134933733, 84656743, 39632149, 13481441, 3156845, 455917, 30649;
...
-
nn = 8; B[n_] := FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]] (1 + y)^Binomial[n, 2]; g[z_] := Sum[(1 + u y)^Binomial[n, 2] z^n/FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]], {n, 0, nn}]; egf[eggf_] := Normal[Series[eggf, {z, 0, nn}]] /.Table[z^i -> z^i*B[i]/i!, {i, 1, nn + 1}]; Map[Drop[#, 1] &, Drop[Map[CoefficientList[#, u] &, Table[n!, {n, 0, nn}]CoefficientList[Series[-Log[egf[1/g[z]]], {z, 0, nn}], z] /. y -> 1], 2]] // Grid (* Geoffrey Critzer, Feb 12 2025 *)
A348902
G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(4*x)).
Original entry on oeis.org
1, 1, 9, 305, 39705, 20412737, 41846783913, 342892875489361, 11236600170415809849, 1472826135905484728387681, 772188014962631262957890704329, 1619397184353040716422147490531778929, 13584491414647344530078887450781292845554521
Offset: 0
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nmax = 12; A[] = 0; Do[A[x] = 1/(1 + x - 2 x A[4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[2^(2 k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]
A348188
G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(3*x)).
Original entry on oeis.org
1, 1, 7, 139, 7813, 1282741, 626077507, 914089078999, 4000061058178633, 52496811551448519241, 2066694521388276020211487, 244076623554395367965602542499, 86475371441574361841467969073397133, 91913288701991663661449175594278601481981
Offset: 0
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nmax = 13; A[] = 0; Do[A[x] = 1/(1 + x - 2 x A[3 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -a[n - 1] + 2 Sum[3^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
A349038
G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(-2*x)).
Original entry on oeis.org
1, 1, -3, -31, 453, 15641, -973443, -126707471, 32192101173, 16547934365321, -16912274385623763, -34670312866958030751, 141940412456349939507813, 1163060052394732038435530361, -19053251054424307861590927924003, -624375047526738670923288994646642991
Offset: 0
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nmax = 15; A[] = 0; Do[A[x] = 1/(1 + x - 2 x A[-2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[(-2)^(k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
A348903
G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x - x * A(2*x)).
Original entry on oeis.org
1, 3, 15, 123, 1623, 35427, 1349727, 94653195, 12690736167, 3325408581747, 1722610175806383, 1774299723226774683, 3644417103927252697335, 14949404433893216347632003, 122555228634241017164802041343, 2008680242472430855727593100321067
Offset: 0
Showing 1-5 of 5 results.
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