cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-28 of 28 results.

A120602 G.f. satisfies: 31*A(x) = 30 + 125*x + A(x)^6, starting with [1,5,15].

Original entry on oeis.org

1, 5, 15, 190, 2550, 38070, 609205, 10199640, 176483340, 3130904150, 56641633455, 1040985874470, 19381240377460, 364777461207360, 6929053224018750, 132665646902812800, 2557591625106894075, 49604907701733017850
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 5*x + 15*x^2 + 190*x^3 + 2550*x^4 + 38070*x^5 +...
A(x)^6 = 1 + 30*x + 465*x^2 + 5890*x^3 + 79050*x^4 + 1180170*x^5 +...

Crossrefs

Cf. A120588 - A120601, A120603 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+31*x - (1+x)^6)/125, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+5*x+15*x^2+x*O(x^n));for(i=0,n,A=A+(-31*A+30+125*x+A^6)/25);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+31*x - (1+x)^6)/125). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (30+125*x)^(5*n+1)/31^(6*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 5^(-1/2 + 3*n) * (-30 + 5*(31/6)^(6/5))^(1/2 - n) / (2^(3/5) * 3^(1/10) * 31^(2/5) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120603 G.f. satisfies: 16*A(x) = 15 + 27*x + A(x)^7, starting with [1,3,21].

Original entry on oeis.org

1, 3, 21, 399, 9135, 233709, 6400947, 183585897, 5443737390, 165536020650, 5133935821014, 161768728483362, 5164132704296202, 166660621950110526, 5428573285691233650, 178234125351736454070, 5892439158797172244515
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 3*x + 21*x^2 + 399*x^3 + 9135*x^4 + 233709*x^5 +...
A(x)^7 = 1 + 21*x + 336*x^2 + 6384*x^3 + 146160*x^4 + 3739344*x^5 +...

Crossrefs

Cf. A120588 - A120602, A120604 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+16*x - (1+x)^7)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+3*x+21*x^2+x*O(x^n));for(i=0,n,A=A+(-16*A+15+27*x+A^7)/9);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+16*x - (1+x)^7)/27). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(7*n,n)/(6*n+1) * (15+27*x)^(6*n+1)/16^(7*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 7^(-13/12 + 2*n) * 9^n * (-245 + 32*2^(2/3)*7^(5/6))^(1/2 - n) / (2^(8/3) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120604 G.f. satisfies: 24*A(x) = 23 + 64*x + A(x)^8, starting with [1,4,28].

Original entry on oeis.org

1, 4, 28, 616, 15820, 453208, 13894552, 445970128, 14796844588, 503423385080, 17467725995720, 615756709476272, 21990183407958584, 793912445913712496, 28928560840589374640, 1062498482335560005024, 39293868860176487815916
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 4*x + 28*x^2 + 616*x^3 + 15820*x^4 + 453208*x^5 +...
A(x)^8 = 1 + 32*x + 672*x^2 + 14784*x^3 + 379680*x^4 + 10876992*x^5 +...

Crossrefs

Cf. A120588 - A120603, A120605 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+24*x - (1+x)^8)/64, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+4*x+28*x^2+x*O(x^n));for(i=0,n,A=A+(-24*A+23+64*x+A^8)/16);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+24*x - (1+x)^8)/64). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(8*n,n)/(7*n+1) * (23+64*x)^(7*n+1)/24^(8*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 4^(-1 + 3*n) * (-23 + 21*3^(1/7))^(1/2 - n) / (3^(3/7) * n^(3/2) * sqrt(7*Pi)). - Vaclav Kotesovec, Nov 28 2017

A120605 G.f. satisfies: 25*A(x) = 24 + 64*x + A(x)^9, starting with [1,4,36].

Original entry on oeis.org

1, 4, 36, 984, 31716, 1140552, 43895208, 1768717872, 73674176868, 3146885203432, 137085166193976, 6066992348458704, 272023207778276136, 12330039492509279184, 564072488005316830416, 26010805156782400648800
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 4*x + 36*x^2 + 984*x^3 + 31716*x^4 + 1140552*x^5 +...
A(x)^9 = 1 + 36*x + 900*x^2 + 24600*x^3 + 792900*x^4 + 28513800*x^5 +...

Crossrefs

Cf. A120588 - A120604, A120606, A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+25*x - (1+x)^9)/64, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+4*x+36*x^2+x*O(x^n));for(i=0,n,A=A+(-25*A+24+64*x+A^9)/16);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+25*x - (1+x)^9)/64). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(9*n,n)/(8*n+1) * (24+64*x)^(8*n+1)/25^(9*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 4^(-1 + 3*n) * (-24 + 8*(5/3)^(9/4))^(1/2 - n) / (3^(1/8) * 5^(7/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120606 G.f. satisfies: 36*A(x) = 35 + 81*x + A(x)^9, starting with [1,3,12].

Original entry on oeis.org

1, 3, 12, 180, 3018, 56238, 1121484, 23406804, 504914175, 11167352013, 251879507880, 5771456609880, 133970974830420, 3143760834627420, 74454455230816008, 1777349666975945784, 42721359085344132657
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 3*x + 12*x^2 + 180*x^3 + 3018*x^4 + 56238*x^5 +...
A(x)^9 = 1 + 27*x + 432*x^2 + 6480*x^3 + 108648*x^4 + 2024568*x^5 +...

Crossrefs

Cf. A120588 - A120605, A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+36*x - (1+x)^9)/81, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+3*x+12*x^2+x*O(x^n));for(i=0,n,A=A+(-36*A+35+81*x+A^9)/27);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+36*x - (1+x)^9)/81). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(9*n,n)/(8*n+1) * (35+81*x)^(8*n+1)/36^(9*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 3^(-1 + 4*n) * (-35 + 2^(21/4))^(1/2 - n) / (2^(23/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A356115 Triangle read by rows. The reduced triangle of the partition triangle of reducible permutations with weakly decreasing Lehmer code (A356266). T(n, k) for n >= 1 and 0 <= k < n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 4, 6, 3, 1, 0, 9, 20, 6, 6, 1, 0, 11, 45, 50, 15, 10, 1, 0, 19, 93, 185, 80, 36, 15, 1, 0, 22, 196, 462, 490, 161, 77, 21, 1, 0, 33, 312, 1120, 1834, 1050, 336, 148, 28, 1
Offset: 1

Views

Author

Peter Luschny, Aug 16 2022

Keywords

Examples

			[ 1] [1]
[ 2] [0,  1]
[ 3] [0,  1,   1]
[ 4] [0,  3,   1,    1]
[ 5] [0,  4,   6,    3,    1]
[ 6] [0,  9,  20,    6,    6,    1]
[ 7] [0, 11,  45,   50,   15,   10,   1]
[ 8] [0, 19,  93,  185,   80,   36,  15,   1]
[ 9] [0, 22, 196,  462,  490,  161,  77,  21,  1]
[10] [0, 33, 312, 1120, 1834, 1050, 336, 148, 28, 1]

Links

Crossrefs

Cf. A356266 (partition version), A356265, A120588 (row sums).

Programs

  • SageMath
    # uses function reduce_partition_triangle from A356265.
def A356115_row(n: int) -> list[int]:
    return reduce_partition_triangle(A356266_row, n + 1)[n - 1]
def A356115(n: int, k: int) -> int:
    return A356115_row(n)[k]
for n in range(1, 11):
    print([n], A356115_row(n))

A356266 Partition triangle read by rows, counting reducible permutations with weakly decreasing Lehmer code, refining triangle A356115.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 3, 3, 1, 0, 1, 4, 4, 2, 6, 12, 2, 4, 6, 1, 0, 1, 5, 5, 5, 10, 20, 10, 10, 10, 30, 10, 5, 10, 1, 0, 1, 6, 6, 6, 3, 15, 30, 30, 15, 15, 20, 60, 30, 60, 5, 15, 60, 30, 6, 15, 1
Offset: 0

Views

Author

Peter Luschny, Aug 16 2022

Keywords

Examples

			[0] 1;
[1] 1;
[2] 0, 1;
[3] 0, 1, 1;
[4] 0, [1, 2], 1, 1;
[5] 0, [1, 3], [3, 3], 3,  1;
[6] 0, [1, 4, 4], [2,  6, 12], [2,  4],  6,  1;
[7] 0, [1, 5, 5], [5, 10, 20, 10], [10, 10, 30], [10,  5], 10,  1;
[8] 0, [1, 6, 6, 6],[3,15, 30, 30, 15],[15, 20, 60, 30, 60],[5,15,60],[30,6],15,1;
Summing the bracketed terms reduces the triangle to A356115.

Links

Crossrefs

Cf. A356264, A356115 (reduced), A120588 (row sums).

Programs

  • SageMath
    # uses functions perm_red_stats and reducible from A356264.
@cache
def A356266_row(n: int) -> list[int]:
    if n < 2: return [1]
    return [0] + [v[1] for v in perm_red_stats(n, reducible, weakly_decreasing)]
def A356266(n: int, k: int) -> int:
    return A356266_row(n)[k]
for n in range(8):
    print(A356266_row(n))

A378145 Riordan triangle (1 + x * C(x), x * C(x)), where C(x) is g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 4, 3, 1, 5, 10, 8, 4, 1, 14, 28, 23, 13, 5, 1, 42, 84, 70, 42, 19, 6, 1, 132, 264, 222, 138, 68, 26, 7, 1, 429, 858, 726, 462, 240, 102, 34, 8, 1, 1430, 2860, 2431, 1573, 847, 385, 145, 43, 9, 1, 4862, 9724, 8294, 5434, 3003, 1430, 583, 198, 53, 10, 1
Offset: 0

Views

Author

Werner Schulte, Nov 17 2024

Keywords

Examples

			Triangle T(n, k) for 0 <= k <= n starts:
n\k :     0     1     2     3    4    5    6   7  8  9
======================================================
  0 :     1
  1 :     1     1
  2 :     1     2     1
  3 :     2     4     3     1
  4 :     5    10     8     4    1
  5 :    14    28    23    13    5    1
  6 :    42    84    70    42   19    6    1
  7 :   132   264   222   138   68   26    7   1
  8 :   429   858   726   462  240  102   34   8  1
  9 :  1430  2860  2431  1573  847  385  145  43  9  1
  etc.

Crossrefs

Cf. A000108, A004070, A120588 (column 0), A068875 (column 1 and row sums), A000007 (alt. row sums).

Programs

  • PARI
    T(n,k)=if(k==n,1,binomial(2*n-k,n)*(n*(3*k+1)-2*k*(k+1))/((2*n-k)*(2*n-k-1)))

Formula

T(n, k) = binomial(2*n-k, n) * (n*(3*k+1) - 2*k*(k+1)) / ((2*n-k) * (2*n-k-1)) if 0 <= k < n and 1 if k = n.
T(n, k) = T(n, k-1) - T(n-1, k-2) for 2 <= k <= n.
(-1)^(n-k) * T(n, k) is matrix inverse of A004070 (seen as a triangle).
Conjecture: Sum_{i=0..n-k} binomial(i+m-1, i) * T(n, i+k) = T(n+m, m+k) for m > 0.
Conjecture: Sum_{k=0..n} (1 + floor(k/2)) * T(n, k) = A000108(n+1).
G.f.: A(x, y) = (1 + x*C(x)) / (1 - y * x*C(x)), where C(x) is g.f. of A000108.
Previous Showing 21-28 of 28 results.

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