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.

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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).

A277310 G.f. satisfies: A(x - 4*A(x)^2) = x - 3*A(x)^2.

Original entry on oeis.org

1, 1, 10, 141, 2422, 47562, 1031764, 24214405, 606444990, 16055089470, 446238074892, 12955112773554, 391332183548956, 12261884937532340, 397576302315045800, 13313017677172350965, 459635990935574444942, 16339309997761322057206, 597340515437542895494748, 22435278085988347895795526, 864900964565994975048855444, 34195693888939483596581262668, 1385553440866978431053220575128
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 10*x^3 + 141*x^4 + 2422*x^5 + 47562*x^6 + 1031764*x^7 + 24214405*x^8 + 606444990*x^9 + 16055089470*x^10 +...
such that A(x - 4*A(x)^2) = x - 3*A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 21*x^4 + 302*x^5 + 5226*x^6 + 102788*x^7 + 2226973*x^8 + 52126582*x^9 + 1301232638*x^10 + 34328704796*x^11 + 950803699394*x^12 + 27510261070028*x^13 + 828332416917876*x^14 + 25876801064095496*x^15 + 836682915170627501*x^16 +...
A(x - 4*A(x)^2) = x - 3*x^2 - 6*x^3 - 63*x^4 - 906*x^5 - 15678*x^6 - 308364*x^7 - 6680919*x^8 - 156379746*x^9 - 3903697914*x^10 +...
which equals x - 3*A(x)^2.
Series_Reversion(x - 4*A(x)^2) = x + 4*x^2 + 40*x^3 + 564*x^4 + 9688*x^5 + 190248*x^6 + 4127056*x^7 + 96857620*x^8 + 2425779960*x^9 + 64220357880*x^10 +...
which equals -3*x + 4*A(x).
A( 4*A(x) - 3*x ) = x + 5*x^2 + 58*x^3 + 921*x^4 + 17494*x^5 + 374994*x^6 + 8793460*x^7 + 221393569*x^8 + 5912166718*x^9 + 166058455158*x^10 + 4876311925036*x^11 + 149037482367530*x^12 + 4724877954111836*x^13 + 154959634972646340*x^14 + 5246331138228520168*x^15 +...
which equals  sqrt( A(x) - x ).

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-4*F^2) + 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + A( 4*A(x) - 3*x )^2.
(2) A(x) = 3*x/4 + 1/4 * Series_Reversion(x - 4*A(x)^2).
(3) R(x) = 4*x/3 - 1/3 * Series_Reversion(x - 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x - R(x) ) ) = 4*x - 3*R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 4^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 4^k.

A276365 G.f. A(x) satisfies: A(x - 2*A(x)^2) = x - A(x)^2.

Original entry on oeis.org

1, 1, 6, 53, 578, 7234, 100044, 1495125, 23802346, 399740086, 7032766196, 128952474242, 2454645604820, 48359400068836, 983683769369624, 20618782389897333, 444636132851851386, 9851377271964349038, 223998085060636514020, 5221799494107885481430, 124695762315403816775932, 3047952254964607540099676, 76206565881709345978097960, 1947752912315470845518308642, 50860833685759573411702643972
Offset: 1

Views

Author

Paul D. Hanna, Sep 01 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 53*x^4 + 578*x^5 + 7234*x^6 + 100044*x^7 + 1495125*x^8 + 23802346*x^9 + 399740086*x^10 + 7032766196*x^11 +...
such that A(x - 2*A(x)^2) = x - A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 2*A(x)^2) = 2*A(x) - x, which begins:
Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1156*x^5 + 14468*x^6 + 200088*x^7 + 2990250*x^8 + 47604692*x^9 + 799480172*x^10 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - x^2 - 4*x^3 - 28*x^4 - 264*x^5 - 2992*x^6 - 38496*x^7 - 544464*x^8 - 8298080*x^9 - 134500672*x^10 - 2297361024*x^11 +...
then Series_Reversion(x - A(x)^2) = 2*x - R(x), and
R(x) = x - G(x)^2, where G(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1208*x^5 + 15536*x^6 + 220832*x^7 + 3390480*x^8 + ... + A177409(n)*x^n + ...
Also, sqrt(A(x) - x) = A(2*A(x) - x), which begins:
sqrt(A(x) - x) = x + 3*x^2 + 22*x^3 + 223*x^4 + 2706*x^5 + 36998*x^6 + 552172*x^7 + 8827263*x^8 + 149328698*x^9 + 2650946274*x^10 + ...

Links

Crossrefs

Cf. A277295, A213591, A275765, A177409.

Programs

  • Mathematica
    m = 26; A[_] = 0;
Do[A[x_] = x + A[2 A[x] - x]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 2*F^2) + F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + A( 2*A(x) - x )^2.
(2) 2*A(x) = x + Series_Reversion(x - 2*A(x)^2).
(3) R(x) = 2*x - Series_Reversion(x - A(x)^2), where R(A(x)) = x.
(4) R( (x - R(x))^(1/2) ) = 2*x - R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 2^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(6) A(x) = x + G(A(x))^2, where G(x) = sqrt(x - R(x)) is the g.f. of A177409, and R(A(x)) = x. - Paul D. Hanna, Nov 18 2022
a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k.

A277311 G.f. satisfies: A(x - 5*A(x)^2) = x - 4*A(x)^2.

Original entry on oeis.org

1, 1, 12, 200, 4034, 92752, 2353272, 64579809, 1891598860, 58591554652, 1906271367296, 64816527248936, 2294331974613872, 84290267670946720, 3206227129084419920, 126022120854865417140, 5110001578581607976400, 213458728365617240931360, 9175021814527973211291880, 405366362599820848509766760, 18392202994173383123235536800, 856255190568423353781484124240
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 12*x^3 + 200*x^4 + 4034*x^5 + 92752*x^6 + 2353272*x^7 + 64579809*x^8 + 1891598860*x^9 + 58591554652*x^10 +...
such that  A(x - 5*A(x)^2) = x - 4*A(x)^2.
A(x)^2 = x^2 + 2*x^3 + 25*x^4 + 424*x^5 + 8612*x^6 + 198372*x^7 + 5028864*x^8 + 137705810*x^9 + 4022209822*x^10 + 124205854376*x^11 + 4028545272136*x^12 + 136566005356212*x^13 + 4820263259998720*x^14 + 176614868022441920*x^15 +...
A(x - 5*A(x)^2) = x - 4*x^2 - 8*x^3 - 100*x^4 - 1696*x^5 - 34448*x^6 - 793488*x^7 - 20115456*x^8 - 550823240*x^9 - 16088839288*x^10 +...
which equals x - 4*A(x)^2.
Series_Reversion(x - 5*A(x)^2) = x + 5*x^2 + 60*x^3 + 1000*x^4 + 20170*x^5 + 463760*x^6 + 11766360*x^7 + 322899045*x^8 + 9457994300*x^9 +...
which equals  5*A(x) - 4*x.
A( 5*A(x) - 4*x ) = x + 6*x^2 + 82*x^3 + 1525*x^4 + 33864*x^5 + 848402*x^6 + 23259832*x^7 + 685028874*x^8 + 21411099560*x^9 + 704295189492*x^10 +24234549363096*x^11 + 868423052983416*x^12 + 32296557071230392*x^13 + 1243216715481216720*x^14 + 49428242214109804120*x^15 +...
which equals  sqrt( A(x) -x ).

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309, A277310.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-5*F^2) + 4*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + A( 5*A(x) - 4*x )^2.
(2) A(x) = 4*x/5 + 1/5 * Series_Reversion(x - 5*A(x)^2).
(3) R(x) = 5*x/4 - 1/4 * Series_Reversion(x - 4*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x - R(x) ) ) = 5*x - 4*R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 5^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 5^k.
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