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.

Showing 1-6 of 6 results.

A291820 G.f. A(x,y) satisfies: A( x - x*y*A(x,y), y) = x + x*(1-y)*A(x,y), where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 7, 5, 0, 1, 16, 38, 14, 0, 1, 30, 157, 189, 42, 0, 1, 50, 477, 1245, 904, 132, 0, 1, 77, 1197, 5616, 8791, 4242, 429, 0, 1, 112, 2632, 19881, 55566, 57854, 19723, 1430, 0, 1, 156, 5250, 59327, 265204, 491947, 363880, 91366, 4862, 0, 1, 210, 9714, 155783, 1035442, 3062271, 4039551, 2220933, 423124, 16796, 0, 1, 275, 16929, 370205, 3472513, 15217674, 31979723, 31463341, 13285415, 1963169, 58786, 0, 1, 352, 28094, 811877, 10331673, 63678254, 197983540, 310618856, 235959185, 78419541, 9138416, 208012, 0
Offset: 1

Views

Author

Paul D. Hanna, Sep 01 2017

Keywords

Comments

More generally, we have the following related identity.
Given functions F and G with F(0)=0, F'(0)=1, G(0)=0, G'(0)=0,
if F(x - y*G(x)) = x + (1-y)*G(x), then
(C1) F(x) = x + G( y*F(x) + (1-y)*x ),
(C2) y*F(x) + (1-y)*x = Series_Reversion(x - y*G(x)),
(C3) F(x) = x + G(x + y*G(x + y*G(x + y*G(x +...)))),
(C4) F(x) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x)^n / n!.
The g.f. A(x,y) of this sequence equals F(x) in the above when G(x) = x*F(x).

Examples

			G.f.: A(x,y)  = x + x^2 + (2*y + 1)*x^3 + (5*y^2 + 7*y + 1)*x^4 +
(14*y^3 + 38*y^2 + 16*y + 1)*x^5 +
(42*y^4 + 189*y^3 + 157*y^2 + 30*y + 1)*x^6 +
(132*y^5 + 904*y^4 + 1245*y^3 + 477*y^2 + 50*y + 1)*x^7 +
(429*y^6 + 4242*y^5 + 8791*y^4 + 5616*y^3 + 1197*y^2 + 77*y + 1)*x^8 +
(1430*y^7 + 19723*y^6 + 57854*y^5 + 55566*y^4 + 19881*y^3 + 2632*y^2 + 112*y + 1)*x^9 +
(4862*y^8 + 91366*y^7 + 363880*y^6 + 491947*y^5 + 265204*y^4 + 59327*y^3 + 5250*y^2 + 156*y + 1)*x^10 +
(16796*y^9 + 423124*y^8 + 2220933*y^7 + 4039551*y^6 + 3062271*y^5 + 1035442*y^4 + 155783*y^3 + 9714*y^2 + 210*y + 1)*x^11 +
(58786*y^10 + 1963169*y^9 + 13285415*y^8 + 31463341*y^7 + 31979723*y^6 + 15217674*y^5 + 3472513*y^4 + 370205*y^3 + 16929*y^2 + 275*y + 1)*x^12 +...
such that
A( x - x*y*A(x,y), y)  =  x + x*(1-y)*A(x,y).
Also,
A(x,y) = x + Z*A(Z, y) where Z = y*A(x,y) + (1-y)*x.
...
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins:
1;
1, 0;
1, 2, 0;
1, 7, 5, 0;
1, 16, 38, 14, 0;
1, 30, 157, 189, 42, 0;
1, 50, 477, 1245, 904, 132, 0;
1, 77, 1197, 5616, 8791, 4242, 429, 0;
1, 112, 2632, 19881, 55566, 57854, 19723, 1430, 0;
1, 156, 5250, 59327, 265204, 491947, 363880, 91366, 4862, 0;
1, 210, 9714, 155783, 1035442, 3062271, 4039551, 2220933, 423124, 16796, 0;
1, 275, 16929, 370205, 3472513, 15217674, 31979723, 31463341, 13285415, 1963169, 58786, 0;
1, 352, 28094, 811877, 10331673, 63678254, 197983540, 310618856, 235959185, 78419541, 9138416, 208012, 0;
1, 442, 44759, 1666522, 27896583, 232505790, 1014785477, 2355151627, 2859824058, 1721756609, 458956233, 42718416, 742900, 0; ...
RELATED SEQUENCES.
Given T(n,k) is the coefficient of x^n*y^k in g.f. A(x,y),
if b(n) = Sum_{k=0..n-1} T(n,k) * p^k * q^(n-k-1)
then B(x) = Sum_{n>=1} b(n)*x^n satisfies
(E1) B(x - p*x*B(x)) = x + (q-p)*x*B(x)
(E2) B(x)  =  x + Z*B(Z) where Z = p*B(x) + (q-p)*x.
...
G.f.s of columns of this triangle begin:
C.0: 1/(1-x)
C.1: (2 - x)/(1-x)^4
C.2: (5 + 3*x - 4*x^2 + x^3)/(1-x)^7
C.3: (14 + 49*x - 15*x^2 - 9*x^3 + 6*x^4 - x^5)/(1-x)^10
C.4: (42 + 358*x + 315*x^2 - 217*x^3 + 30*x^4 + 18*x^5 - 8*x^6 + x^7)/(1-x)^13
C.5: (132 + 2130*x + 5822*x^2 + 1403*x^3 - 1681*x^4 + 602*x^5 - 50*x^6 - 30*x^7 + 10*x^8 - x^9)/(1-x)^16
C.6: (429 + 11572*x + 62502*x^2 + 82763*x^3 + 2951*x^4 - 9760*x^5 + 5395*x^6 - 1329*x^7 + 75*x^8 + 45*x^9 - 12*x^10 + x^11)/(1-x)^19
C.7: (1430 + 59906*x + 541211*x^2 + 1506161*x^3 + 1217687*x^4 + 16416*x^5 - 35746*x^6 + 36682*x^7 - 13502*x^8 + 2550*x^9 - 105*x^10 - 63*x^11 + 14*x^12 - x^13)/(1-x)^22
C.8: (4862 + 301574*x + 4165915*x^2 + 19578410*x^3 + 34788033*x^4 + 20899306*x^5 + 1681742*x^6 + 174039*x^7 + 195964*x^8 - 103084*x^9 + 28953*x^10 - 4444*x^11 + 140*x^12 + 84*x^13 - 16*x^14 + x^15)/(1-x)^25
...
Thus A(x, y*(1-x)^3)*(1-x) = x + 2*y*x^3 + (5*y^2 - y)*x^4 + (14*y^3 + 3*y^2)*x^5 + (42*y^4 + 49*y^3 - 4*y^2)*x^6 + (132*y^5 + 358*y^4 - 15*y^3 + y^2)*x^7 +...

Links

Crossrefs

Cf. A088714 (row sums), A291821 (central terms), A291822 (diagonal).
Cf. A291813, A291814, A291815, A291816, A291817, A291818, A291819.
Cf. A276358, A291743, A291744.
Cf. A277295 (variant).

Programs

  • Mathematica
    nmax = 13; A[x_] = x;
Do[A[x_] = x + (y A[x] + (1-y) x) A[y A[x] + (1-y) x] + x O[x]^n // Normal // Expand // Collect[#, x]&, {n, nmax}];
T[n_, k_] := SeriesCoefficient[A[x], {x, 0, n}, {y, 0, k}];
Table[T[n, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Oct 20 2019 *)
  • PARI
    {T(n, k) = my(A=x); for(i=1, n, A = x + subst(x*A, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A, n, x), k, y)}
for(n=1, 12, for(k=0, n-1, print1(T(n, k), ", ")); print(""))

Formula

G.f. A(x,y) also satisfies:
(G1) A(x,y) = x + A( y*A(x,y) + x*(1-y), y).
(G2) y*A(x,y) + x*(1-y) = Series_Reversion( x - x*y*A(x,y) ).
(G3) x*y + (1-y)*B(x,y) = Series_Reversion( x + x*(1-y)*A(x,y) ), where B( A(x,y), y) = x.
(G4) A(x,y) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) A(x,y)^n * x^n / n!.
In formulas 2 and 3, the series reversion is taken with respect to variable x.
Formulas for terms:
(T1) T(n,0) = 1.
(T2) T(n,1) = (n-1)*n*(n+4)/6. for n>=1.
(T3) T(n+1,n-1) = binomial(2*n,n)/(n+1) = A000108(n) for n>=1.
Row sums:
(S1) Sum_{k=0..n-1} T(n,k) = A088714(n-1).
(S2) Sum_{k=0..n-1} T(n,k) * 2^(n-k-1) = A276358(n).
(S3) Sum_{k=0..n-1} T(n,k) * 3^(n-k-1) = A291744(n).
(S4) Sum_{k=0..n-1} T(n,k) * 2^k * 3^(n-k-1) = A291743(n).
(S5) Sum_{k=0..n-1} T(n,k) * 2^k = A291813(n).
(S6) Sum_{k=0..n-1} T(n,k) * 3^k = A291814(n).
(S7) Sum_{k=0..n-1} T(n,k) * 4^k = A291815(n).
(S8) Sum_{k=0..n-1} T(n,k) * 3^k * 2^(n-k-1) = A291816(n).
(S9) Sum_{k=0..n-1} T(n,k) * 3^k * 4^(n-k-1) = A291817(n).
(S10) Sum_{k=0..n-1} T(n,k) * 4^k * 3^(n-k-1) = A291818(n).
(S11) Sum_{k=0..n-1} T(n,k) * 4^(n-k-1) = A291819(n).

A291744 G.f. A(x) satisfies: A(x - x*A(x)) = x + 2*x*A(x).

Original entry on oeis.org

1, 3, 15, 105, 897, 8739, 93663, 1080909, 13246017, 170728251, 2298619851, 32162768805, 465875706873, 6964550221215, 107193366978651, 1695277029466917, 27504875620268325, 457183442035485927, 7776605660061178251, 135234473290510961097, 2402252449086179775861, 43557766261735276367055, 805650777590230815177879, 15191845940176304945626737, 291896599103455803872483709, 5712079123789080942126760083
Offset: 1

Views

Author

Paul D. Hanna, Aug 30 2017

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 15*x^3 + 105*x^4 + 897*x^5 + 8739*x^6 + 93663*x^7 + 1080909*x^8 + 13246017*x^9 + 170728251*x^10 + 2298619851*x^11 + 32162768805*x^12 +...
such that A(x - x*A(x)) = x + 2*x*A(x).
RELATED SERIES.
A(x - x*A(x)) = x + 2*x^2 + 6*x^3 + 30*x^4 + 210*x^5 + 1794*x^6 + 17478*x^7 + 187326*x^8 + 2161818*x^9 + 26492034*x^10 + 341456502*x^11 + 4597239702*x^12 +...
which equals x + 2*x*A(x).
Series_Reversion( x - x*A(x) ) = x + x^2 + 5*x^3 + 35*x^4 + 299*x^5 + 2913*x^6 + 31221*x^7 + 360303*x^8 + 4415339*x^9 + 56909417*x^10 + 766206617*x^11 + 10720922935*x^12 +...
which equals (2*x + A(x))/3.
A( (2*x + A(x))/3 ) = x + 4*x^2 + 26*x^3 + 218*x^4 + 2126*x^5 + 22986*x^6 + 268410*x^7 + 3331482*x^8 + 43492370*x^9 + 592851806*x^10 + 8393229602*x^11 + 122922601030*x^12 +...
which equals (A(x) - x) / (A(x) + 2*x).

Links

Crossrefs

Cf. A291743, A276358.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = 3*serreverse( x - x*A +x*O(x^n) ) - 2*x ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=x, B); for(i=1, n, B = (2*x + A)/3 +x*O(x^n); A = x*(1 + 2*subst(A, x, B))/(1 - subst(A, x, B)) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = 3 * Series_Reversion( x - x*A(x) ) - 2*x.
(2) A(x) = x * (1 + 2*A(B(x))) / (1 - A(B(x))), where B(x) = (2*x + A(x))/3.
(3) A( (2*x + A(x))/3 ) = (A(x) - x) / (A(x) + 2*x).

A291813 G.f. A(x) satisfies: A(x - 2*x*A(x)) = x - x*A(x).

Original entry on oeis.org

1, 1, 5, 35, 297, 2873, 30657, 353727, 4355497, 56709337, 775575269, 11085971235, 164979882033, 2548461481105, 40762085472929, 673751263927071, 11489101983573105, 201838769635965969, 3648620371959258149, 67795012307507004291, 1293607920940368319641, 25326486746707799668105, 508368313083167614599201, 10454499119633293760277151, 220120546753823908307191769, 4742197866143368618862457641
Offset: 1

Views

Author

Paul D. Hanna, Sep 01 2017

Keywords

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 35*x^4 + 297*x^5 + 2873*x^6 + 30657*x^7 + 353727*x^8 + 4355497*x^9 + 56709337*x^10 + 775575269*x^11 + 11085971235*x^12 +...
such that  A(x - 2*x*A(x)) = x - x*A(x).
RELATED SERIES.
A(x - 2*x*A(x)) = x - x^2 - x^3 - 5*x^4 - 35*x^5 - 297*x^6 - 2873*x^7 - 30657*x^8 +...
which equals x - x*A(x).
Series_Reversion( x - 2*x*A(x) ) = x + 2*x^2 + 10*x^3 + 70*x^4 + 594*x^5 + 5746*x^6 + 61314*x^7 + 707454*x^8 + 8710994*x^9 + 113418674*x^10 +...
which equals 2*A(x) - x.
A( 2*A(x) - x ) = x + 3*x^2 + 19*x^3 + 159*x^4 + 1561*x^5 + 17087*x^6 + 202975*x^7 + 2574391*x^8 + 34495545*x^9 + 484770627*x^10 + 7107406323*x^11 + 108289787415*x^12 + 1709478736593*x^13 + 27894511442079*x^14 +...
which equals (A(x) - x) / (2*A(x) - x).

Crossrefs

Cf. A291820, A291814, A291815, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = (1/2)*serreverse( x - 2*x*A +x*O(x^n) ) + x/2 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = (1/2)*Series_Reversion( x - 2*x*A(x) ) + x/2.
(2) A( 2*A(x) - x ) = (A(x) - x) / (2*A(x) - x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 2^k.

A291814 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x - 2*x*A(x).

Original entry on oeis.org

1, 1, 7, 67, 769, 10009, 143359, 2218255, 36625657, 639659737, 11741022235, 225390779647, 4508109360985, 93665093491381, 2016669357747667, 44905700922069463, 1032419000661778213, 24472819932819733957, 597384952530618840715, 15000294032677574361955, 387082666821619977435277, 10256260095368150955828565, 278811213889895147327704519, 7770474960716476086765483619
Offset: 1

Views

Author

Paul D. Hanna, Sep 02 2017

Keywords

Examples

			G.f.: A(x) = x + x^2 + 7*x^3 + 67*x^4 + 769*x^5 + 10009*x^6 + 143359*x^7 + 2218255*x^8 + 36625657*x^9 + 639659737*x^10 + 11741022235*x^11 + 225390779647*x^12 +...
such that  A(x - 3*x*A(x)) = x - 2*x*A(x).
RELATED SERIES.
A(x - 3*x*A(x)) = x - 2*x^2 - 2*x^3 - 14*x^4 - 134*x^5 - 1538*x^6 - 20018*x^7 +...
which equals x - 2*x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 21*x^3 + 201*x^4 + 2307*x^5 + 30027*x^6 + 430077*x^7 + 6654765*x^8 +...
which equals 3*A(x) - 2*x.
A( 3*A(x) - 2*x ) = x + 4*x^2 + 34*x^3 + 382*x^4 + 5038*x^5 + 74134*x^6 + 1184650*x^7 + 20224990*x^8 + 364994554*x^9 + 6911857450*x^10 + 136622440786*x^11 + 2807805653098*x^12 +...
which equals (A(x) - x) / (3*A(x) - 2*x).

Crossrefs

Cf. A291820, A291813, A291815, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = (1/3)*serreverse( x - 3*x*A +x*O(x^n) ) + 2*x/3 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = (1/3)*Series_Reversion( x - 3*x*A(x) ) + 2*x/3.
(2) A( 3*A(x) - 2*x ) = (A(x) - x) / (3*A(x) - 2*x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k.

A291815 G.f. A(x) satisfies: A(x - 4*x*A(x)) = x - 3*x*A(x).

Original entry on oeis.org

1, 1, 9, 109, 1569, 25481, 454105, 8730373, 178996865, 3881556561, 88477557289, 2109927671453, 52443846331297, 1354646602217945, 36275862587452281, 1005099719255707829, 28765965099599741953, 849204340574458575777, 25827102287376124267593, 808349897942417046805197, 26011340193853765710238241, 859773626049480606121078057, 29168437337569276216572259097
Offset: 1

Views

Author

Paul D. Hanna, Sep 02 2017

Keywords

Examples

			G.f.: A(x) = x + x^2 + 9*x^3 + 109*x^4 + 1569*x^5 + 25481*x^6 + 454105*x^7 + 8730373*x^8 + 178996865*x^9 + 3881556561*x^10 + 88477557289*x^11 + 2109927671453*x^12 +...
such that  A(x - 4*x*A(x)) = x - 3*x*A(x).
RELATED SERIES.
A(x - 4*x*A(x)) = x - 3*x^2 - 3*x^3 - 27*x^4 - 327*x^5 - 4707*x^6 - 76443*x^7 +...
which equals x - 3*x*A(x).
Series_Reversion( x - 4*x*A(x) ) = x + 4*x^2 + 36*x^3 + 436*x^4 + 6276*x^5 + 101924*x^6 + 1816420*x^7 + 34921492*x^8 +...
which equals 4*A(x) - 3*x.
A( 4*A(x) - 3*x )  = x + 5*x^2 + 53*x^3 + 741*x^4 + 12153*x^5 + 222405*x^6 + 4421501*x^7 + 93949493*x^8 + 2110952881*x^9 + 49786323589*x^10 + 1225967873349*x^11 + 31395927333829*x^12 +...
which equals (A(x) - x) / (4*A(x) - 3*x).

Crossrefs

Cf. A291820, A291813, A291814, A291816, A291817, A291818, A291819, A276358, A291743, A291744.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = (1/4)*serreverse( x - 4*x*A +x*O(x^n) ) + 3*x/4 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = (1/4)*Series_Reversion( x - 4*x*A(x) ) + 3*x/4.
(2) A( 4*A(x) - 3*x) = (A(x) - x) / (4*A(x) - 3*x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 4^k.

A291816 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x - x*A(x).

Original entry on oeis.org

1, 2, 16, 182, 2524, 39992, 699520, 13231034, 266985280, 5694001172, 127481465536, 2981125793144, 72532301230672, 1830526849868000, 47802726801684544, 1289123410465365782, 35841130838977837348, 1025903099063974343984, 30195807234087904770952, 912951678159786641659796, 28327442752528049524839856, 901289532361030971832330544, 29382621186595702051011638128
Offset: 1

Views

Author

Paul D. Hanna, Sep 02 2017

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 16*x^3 + 182*x^4 + 2524*x^5 + 39992*x^6 + 699520*x^7 + 13231034*x^8 + 266985280*x^9 + 5694001172*x^10 +...
such that  A(x - 3*x*A(x)) = x - x*A(x).
RELATED SERIES.
A(x - 3*x*A(x)) = x - x^2 - 2*x^3 - 16*x^4 - 182*x^5 - 2524*x^6 - 39992*x^7 - 699520*x^8 +...
which equals x - x*A(x).
Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 24*x^3 + 273*x^4 + 3786*x^5 + 59988*x^6 + 1049280*x^7 + 19846551*x^8 +...
which equals (3/2)*A(x) - x/2.
A( (3*A(x) - x)/2 )  = x + 5*x^2 + 52*x^3 + 713*x^4 + 11458*x^5 + 205160*x^6 + 3984304*x^7 + 82576109*x^8 + 1807215616*x^9 + 41461917398*x^10 +...
which equals (A(x) - x) / (3*A(x) - x).

Crossrefs

Cf. A291820, A291813, A291814, A291815, A291817, A291818, A291819, A276358, A291743, A291744.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = (2/3)*serreverse( x - 3*x*A +x*O(x^n) ) + x/3 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = (2/3)*Series_Reversion( x - 3*x*A(x) ) + x/3.
(2) A( (3*A(x) - x)/2 ) = (A(x) - x) / (3*A(x) - x).
a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k * 2^(n-k-1).
Showing 1-6 of 6 results.

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