A075834 -id:A075834 - cp's OEIS frontend

A386449 G.f. A(x) satisfies A(x) = 1/(1 - x - x^4*A'''(x)).

1, 1, 1, 1, 7, 181, 11215, 1368049, 290015209, 98023774645, 49599740115757, 35810914359761065, 35524377449180431975, 46963191178201310535625, 80682726920407341929523811, 176372394085267937467487988481, 481849299958664384125278899595601, 1619977170089211596368385150640702601

Offset: 0

Seiichi Manyama, Jul 22 2025

nonn

Cf. A075834, A386448, A386450, A386451.

Cf. A385763.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, sum(k=1, 3, stirling(3, k, 1)*j^k)*v[j+1]*v[i-j])); v;

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (2*k - 3*k^2 + k^3) * a(k) * a(n-1-k).

A386450 G.f. A(x) satisfies A(x) = 1/(1 - x - x^5*A''''(x)).

Original entry on oeis.org

1, 1, 1, 1, 1, 25, 3049, 1103713, 929323297, 1563120681841, 4730002253928145, 23848669801185169825, 188929157434986723256801, 2244856224793842495701519113, 38526222340982767558002054899641, 925631015719595748793089592291450945, 30325523298479173582153602405524578371265

Offset: 0

Seiichi Manyama, Jul 22 2025

nonn

Cf. A075834, A386448, A386449, A386451.

Cf. A385764.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, sum(k=1, 4, stirling(4, k, 1)*j^k)*v[j+1]*v[i-j])); v;

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (-6*k + 11*k^2 - 6*k^3 + k^4) * a(k) * a(n-1-k).

A386451 G.f. A(x) satisfies A(x) = 1/(1 - x - x^6*A'''''(x)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 121, 87361, 220324321, 1481019998401, 22395984195495601, 677299352559157967041, 37550830682188851813205921, 3568906049019293501471580099841, 551188987985086896272084982413188201, 132418744847944340085178947237195978556801, 47718683730343729293790168893699493431209021761

Offset: 0

Seiichi Manyama, Jul 22 2025

nonn

Cf. A075834, A386448, A386449, A386450.

Cf. A385765.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, sum(k=1, 5, stirling(5, k, 1)*j^k)*v[j+1]*v[i-j])); v;

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (24*k - 50*k^2 + 35*k^3 - 10*k^4 + k^5) * a(k) * a(n-1-k).

A386453 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(k+2,3) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 11, 131, 2888, 107027, 6212005, 534389458, 65203760863, 10889677250198, 2417582805875622, 696275799766601842, 254839529849806176727, 116462397939843834894367, 65452132793842930368844779, 44638474752168615525812508053, 36514339485766910607857620043816

Offset: 0

Seiichi Manyama, Jul 22 2025

nonn

Cf. A075834, A386452, A386454, A386455.

Cf. A385875.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, binomial(j+2, 3)*v[j+1]*v[i-j])); v;

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..3} binomial(2,k-1) * x^k/k! * (d^k/dx^k A(x)) ).

A386454 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(k+3,4) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 13, 220, 8148, 586948, 75141039, 15930666825, 5289069956220, 2628685323745449, 1884772989271329869, 1890430039448133854031, 2584219798288871040676608, 4708450397910844142927823544, 11215531466814325127916787062534, 34341962107081618846057340207455738

Offset: 0

Seiichi Manyama, Jul 22 2025

nonn

Cf. A075834, A386452, A386453, A386455.

Cf. A385876.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, binomial(j+3, 4)*v[j+1]*v[i-j])); v;

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..4} binomial(3,k-1) * x^k/k! * (d^k/dx^k A(x)) ).

A386455 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(k+4,5) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 15, 344, 19962, 2555592, 649147331, 301207446317, 239159429472132, 308276821981867349, 617786997525975886618, 1856450241316927094671750, 8112688179283378712969957414, 50217541700003149682333160103969, 430364340522944093019900101527085125

Offset: 0

Seiichi Manyama, Jul 22 2025

nonn

Cf. A075834, A386452, A386453, A386454.

Cf. A385877.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, binomial(j+4, 5)*v[j+1]*v[i-j])); v;

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..5} binomial(4,k-1) * x^k/k! * (d^k/dx^k A(x)) ).

A091063 Triangle, read by rows, such that the initial terms of the binomial transform of the n-th row forms the n-th row of triangle A059438 transposed (permutations of [1..n] with k components).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 7, 0, 1, 4, 9, 18, 34, 0, 1, 5, 14, 34, 86, 206, 0, 1, 6, 20, 56, 162, 508, 1476, 0, 1, 7, 27, 85, 269, 939, 3549, 12123, 0, 1, 8, 35, 122, 415, 1540, 6413, 28498, 111866, 0, 1, 9, 44, 168, 609, 2361, 10314, 50382, 257922, 1143554, 0, 1

Offset: 0

Paul D. Hanna, Dec 17 2003

The main diagonal equals A075834 shift 1 place left; subsequent diagonals of this triangle are self-convolutions of the main diagonal. A075834 has the property that the n-th term of the n-th self-convolution of A075834 equals n!. The first (n+1) terms of the binomial transform of the n-th row forms the n-th row of triangle A059438 transposed, which has row sums equal to the factorials. A059438 is also formed from the self-convolutions of its main diagonal (A003319).

			Rows begin:
{1},
{1,0},
{1,1,0},
{1,2,2,0},
{1,3,5,7,0},
{1,4,9,18,34,0},
{1,5,14,34,86,206,0},
{1,6,20,56,162,508,1476,0},
{1,7,27,85,269,939,3549,12123,0},...
Initial terms of the binomial transform of each row forms A059438:
{1},
{1,1},
{1,2,3},
{1,3,7,13},
{1,4,12,32,71},
{1,5,18,58,177,461},
{1,6,25,92,327,1142,3447},
{1,7,33,135,531,2109,8411,29093},
{1,8,42,188,800,3440,15366,69692,273343},...
which has row sums equal to the factorials.

Cf. A075834, A059438, A003319.

A136633 G.f.: A(x) = Series_Reversion( x / Sum_{n>=0} (n+1)!*x^n ).

Original entry on oeis.org

1, 2, 10, 68, 544, 4832, 46312, 471536, 5055328, 56795840, 667286656, 8197599104, 105446118784, 1423627264256, 20234885027968, 303737480337152, 4827671316780544, 81385455480335360, 1455806861755411456

Offset: 0

Paul D. Hanna, Jan 14 2008

nonn

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 68*x^3 + 544*x^4 + 4832*x^5 + 46312*x^6 +...
Let F(x) = 1 + 2x + 6x^2 + 24x^3 + 120x^4 + 720x^5 +...+ (n+1)!*x^n +...
then A(x) = F(x*A(x)) and A(x/F(x)) = F(x);
also, a(n) = coefficient of x^n in F(x)^n divided by (n+1).
The g.f. A(x) also satisfies:
A(x) = 1 + 2*x*A(x)/(1+x*A(x)) + 2*2^2*x^2*A(x)^2/(1+2*x*A(x))^2 + 2*3^3*x^3*A(x)^3/(1+3*x*A(x))^3 + 2*4^4*x^4*A(x)^4/(1+4*x*A(x))^4 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..446

Cf. A090753, A075834, A211207.

a(n)=polcoeff(serreverse(x/sum(k=0,n,(k+1)!*x^k +x*O(x^n))),n)

a(n)=local(A=1+x); for(i=1, n, A=1+2*sum(m=1, n, m^m*x^m*A^m/(1+m*x*A+x*O(x^n))^m)); polcoeff(A, n)
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 04 2013

G.f. A(x) satisfies:
(1) A(x) = F(x*A(x)) and A(x/F(x)) = F(x);
(2) a(n) = [x^n] F(x)^n / (n+1);
where F(x) = Sum_{n>=0} (n+1)!*x^n.
G.f. satisfies: A(x) = 1 + 2*Sum_{n>=1} n^n * x^n * A(x)^n / (1 + n*x*A(x))^n. - Paul D. Hanna, Feb 04 2013
a(n) ~ exp(2) * n! * n. - Vaclav Kotesovec, Nov 23 2024

A230253 Coefficients of power series A(x) such that coefficient of x^n in A(x)^(n+1) equals (n+1)*(n+2)!/2 for n>=0.

Original entry on oeis.org

1, 3, 3, 6, 27, 162, 1206, 10476, 103059, 1125738, 13473378, 174997908, 2448791838, 36706645908, 586646510796, 9957100024152, 178868488496643, 3390603439026618, 67639341903290730, 1416612563019545220, 31079692422132040170, 712855563504590236860, 17061654943814209044660

Offset: 0

Paul D. Hanna, Oct 13 2013

nonn

			G.f. A(x) = 1 + 3*x + 3*x^2 + 6*x^3 + 27*x^4 + 162*x^5 + 1206*x^6 +...
The coefficients in A(x)^n begin:
n=1: [(1),3,   3,    6,    27,    162,   1206,   10476,   103059, ...];
n=2: [1, (6), 15,   30,    99,    522,   3582,   29484,   278883, ...];
n=3: [1,  9, (36),  99,   297,   1323,   8208,   63342,   572751, ...];
n=4: [1, 12,  66, (240),  783,   3132,  17298,  123552,  1060155, ...];
n=5: [1, 15, 105,  480, (1800),  7083,  35415,  231660,  1869885, ...];
n=6: [1, 18, 153,  846,  3672, (15120), 71415,  428490,  3226797, ...];
n=7: [1, 21, 210, 1365,  6804,  30240,(141120), 789939,  5529573, ...];
n=8: [1, 24, 276, 2064, 11682,  56736, 270720,(1451520), 9485343, ...];
n=9: [1, 27, 351, 2970, 18873, 100440, 500904, 2643840,(16329600), ...]; ...
where the coefficient of x^n in A(x)^(n+1) equals (n+1)*(n+2)!/2 for n>=0.
Notice that the diagonal above the main diagonal forms A111546.

Cf. A075834, A090753, A111546.

{a(n)=polcoeff(x/serreverse(sum(m=1, n+1, (m+1)!/2*x^m)+x^2*O(x^n)), n)}

[x^n] A(x)^n = A111546(n) for n>=0.

A363431 Number of 123-avoiding stabilized-interval-free permutations of size n.

Original entry on oeis.org

1, 1, 1, 2, 5, 14, 44, 150, 496, 1758, 6018, 21782, 76414, 280448, 1001752, 3714032, 13450270, 50259604, 183995056, 691863078, 2555043320, 9657267848, 35921300392, 136360740016, 510267869416, 1944193285228, 7312488701868, 27950641500876, 105590010259396, 404724123141348, 1534775681029994

Offset: 0

Juan B. Gil, Jun 22 2023

nonn

A stabilized-interval-free (SIF) permutation on [n] = {1, 2, ..., n} is one that does not stabilize any proper subinterval of [n].

			For n=4 the a(4)=5 permutations are 2413, 3142, 3412, 3421, 4312.

Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.

Cf. A075834.

For n>2, a(n) = f_0(n) - f_1(n-1) + f_2(n) - Sum_{k=1..floor((n-3)/2)} C(k)^2*a(n-2*k), where C(k)=binomial(2*k,k)/(k+1) and f_j(m) denotes the number of 123-avoiding permutations of size m having j fixed points.

cp's OEIS Frontend

A386449 G.f. A(x) satisfies A(x) = 1/(1 - x - x^4*A'''(x)).

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A386450 G.f. A(x) satisfies A(x) = 1/(1 - x - x^5*A''''(x)).

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A386451 G.f. A(x) satisfies A(x) = 1/(1 - x - x^6*A'''''(x)).

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A386453 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(k+2,3) * a(k) * a(n-1-k).

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A386454 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(k+3,4) * a(k) * a(n-1-k).

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A386455 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(k+4,5) * a(k) * a(n-1-k).

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A091063 Triangle, read by rows, such that the initial terms of the binomial transform of the n-th row forms the n-th row of triangle A059438 transposed (permutations of [1..n] with k components).

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A136633 G.f.: A(x) = Series_Reversion( x / Sum_{n>=0} (n+1)!*x^n ).

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A230253 Coefficients of power series A(x) such that coefficient of x^n in A(x)^(n+1) equals (n+1)*(n+2)!/2 for n>=0.

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A363431 Number of 123-avoiding stabilized-interval-free permutations of size n.

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