A211789 -id:A211789 - cp's OEIS frontend

A369012 Expansion of (1/x) * Series_Reversion( x * (1-x/(1-x))^3 ).

1, 3, 18, 133, 1095, 9636, 88718, 843993, 8230671, 81841987, 826641816, 8457710604, 87472494564, 912995025912, 9604763388534, 101736967518497, 1084125909550959, 11614159795566489, 125011746270524690, 1351312626871871661, 14662950224977228047

Offset: 0

Seiichi Manyama, Jan 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..941
Index entries for reversions of series

Cf. A369013, A369014.
Cf. A243659, A378612.
Cf. A001003, A211789, A378610.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x))^3)/x)

a(n, s=1, t=3, u=-3) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(n-1,n-k).
D-finite with recurrence 96*(3*n+2)*(3*n+1)*(n+1)*a(n) +4*(-4121*n^3 +1922*n^2 -1273*n+124)*a(n-1) +4*(20588*n^3 -76648*n^2 +98677*n -43586)*a(n-2) +(-90073*n^3 +671565*n^2 -1665278*n +1375320)*a(n-3) +210*(n-4)*(3*n-7) *(3*n-8)*a(n-4)=0. - R. J. Mathar, Jan 25 2024
From Seiichi Manyama, Dec 02 2024: (Start)
G.f.: exp( Sum_{k>=1} A378612(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x))^(3*(n+1)).
G.f.: B(x)^3 where B(x) is the g.f. of A243659.
a(n) = 3 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(3*n+k+3,n)/(3*n+k+3). (End)

A286785 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 2, 5, 2, 14, 14, 2, 42, 72, 27, 2, 132, 330, 220, 44, 2, 429, 1430, 1430, 520, 65, 2, 1430, 6006, 8190, 4550, 1050, 90, 2, 4862, 24752, 43316, 33320, 11900, 1904, 119, 2, 16796, 100776, 217056, 217056, 108528, 27132, 3192, 152, 2, 58786, 406980, 1046520, 1302336, 854658, 301644, 55860, 5040, 189, 2, 208012, 1634380, 4903140, 7354710, 6056820, 2826516, 743820, 106260, 7590, 230, 2

Offset: 0

Gheorghe Coserea, May 15 2017

Row n>0 contains n terms.

T(n,k) is the number of Feynman's diagrams with k fermionic loops in the order n of the perturbative expansion in dimension zero for the GW approximation of the polarization function in a many-body theory of fermions with two-body interaction (see Molinari link).

			A(x;t) = 1 + 2*x + (5 + 2*t)*x^2 + (14 + 14*t + 2*t^2)*x^3 + ...
Triangle starts:
   n\k |     0       1       2       3       4      5     6    7  8
  -----+-----------------------------------------------------------
   0   |     1;
   1   |     2;
   2   |     5,      2;
   3   |    14,     14,      2;
   4   |    42,     72,     27,      2;
   5   |   132,    330,    220,     44,      2;
   6   |   429,   1430,   1430,    520,     65,     2;
   7   |  1430,   6006,   8190,   4550,   1050,    90,    2;
   8   |  4862,  24752,  43316,  33320,  11900,  1904,  119,   2;
   9   | 16796, 100776, 217056, 217056, 108528, 27132, 3192, 152, 2;

Gheorghe Coserea, Rows n = 0..123, flattened

Cf. A000108, A286781, A286782, A286783.

T(n,k):=(binomial(n-1,k)*binomial(2*(n+1),n-k))/(n+1); /* Vladimir Kruchinin, Jan 14 2022 */

A286784_ser(N,t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
A286785_ser(N,t='t) = 1/(1-x*A286784_ser(N,t))^2;
concat(apply(p->Vecrev(p), Vec(A286785_ser(12))))

y(x;t) = Sum_{n>=0} P_n(t)*x^n = 1/(1-x*s)^2, where s(x;t) = A286784(x;t) and P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.
A000108(n+1) = T(n,0), A002058(n+3) = T(n,1), A014106(n-1) = T(n,n-2), A006013(n) = P_n(1), A211789(n+1) = P_n(2).
T(n,k) = C(n-1,k)*C(2*n+2,n-k)/(n+1). - Vladimir Kruchinin, Jan 14 2022

A369011 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^2 ).

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 17, 36, 59, 240, 669, 1452, 4538, 13574, 34505, 99816, 299112, 825768, 2364715, 7023466, 20182611, 58327250, 172491553, 505163444, 1476966513, 4370772096, 12924382671, 38149522136, 113266357609, 336894290910, 1001473479313, 2985508193930

Offset: 0

Seiichi Manyama, Jan 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A211789, A368957.
Cf. A054514, A369014.
Cf. A368962, A368966, A368968.

my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x^3/(1-x))^2)/x)

a(n, s=3, t=2, u=-2) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

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

A368957 Expansion of (1/x) * Series_Reversion( x * (1-x^2/(1-x))^2 ).

Original entry on oeis.org

1, 0, 2, 2, 13, 28, 127, 376, 1522, 5210, 20403, 74952, 292313, 1114704, 4371839, 17040586, 67378981, 266402370, 1061919289, 4241539218, 17030430061, 68554148388, 276988107861, 1121954081852, 4557637048543, 18556386241468, 75729621399950

Offset: 0

Seiichi Manyama, Jan 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A211789, A369011.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x^2/(1-x))^2)/x)

a(n, s=2, t=2, u=-2) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(n-k-1,n-2*k).

D-finite with recurrence 2000*n*(48911424697856946605*n -85862091501967897127)*(2*n+1) *(2*n-1)*(n+1)*a(n) +20*n*(2*n-1) *(9782284939571389321000*n^3 -124853950521493511435497*n^2 +291346534864358121613940*n -174094174192357320452243)*a(n-1) +6*(-1056620466555214160730036*n^5 +5240184994626612582867927*n^4 -10842595636486250859803566*n^3 +12555800263623324081669713*n^2 -8323849827256795107408998*n +2408908212964334471344960)*a(n-2) +(-11765946248792268093670721*n^5 +111908835475719217483707009*n^4 -409273054609037480568616913*n^3 +706828511197147489881004671*n^2 -556026097737885029117618846*n +145005575225258917734060720)*a(n-3) +12*(110108843793156901781209*n^5 -1706708924562157727758594*n^4 +10728825545391547292463142*n^3 -34121900584137543620498771*n^2+54762746448568812780284884*n -35381689886652975706836240)*a(n-4) -36*(3*n-11)*(n-4)*(3*n-13) *(2*n-7)*(36626509829570139536*n -97211536327074911575)*a(n-5)=0. - R. J. Mathar, Jan 25 2024

A378610 Expansion of (1/x) * Series_Reversion( x * (1 - x/(1 - x))^4 ).

Original entry on oeis.org

1, 4, 30, 276, 2825, 30884, 353108, 4170500, 50485764, 623084056, 7810707894, 99175174284, 1272856327470, 16486135484248, 215212582153840, 2828658852385572, 37401956484705132, 497174193516767600, 6640063367021736728, 89058042321373540912, 1199031374607501831273

Offset: 0

Seiichi Manyama, Dec 01 2024

nonn

Index entries for reversions of series

Cf. A001003, A211789, A369012.

Cf. A243667, A371486, A378613.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x))^4)/x)

a(n, s=1, t=4, u=-4) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

G.f.: exp( Sum_{k>=1} A378613(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x))^(4*(n+1)).
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(n-1,n-k).
G.f.: B(x)^4 where B(x) is the g.f. of A243667.
a(n) = 4 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(4*n+k+4,n)/(4*n+k+4).

A211788 Triangle enumerating certain two-line arrays of positive integers.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 7, 21, 21, 1, 10, 47, 126, 126, 1, 13, 82, 324, 818, 818, 1, 16, 126, 642, 2300, 5594, 5594, 1, 19, 179, 1107, 4977, 16741, 39693, 39693, 1, 22, 241, 1746, 9335, 38642, 124383, 289510, 289510, 1, 25, 312, 2586, 15941, 77273, 301630, 939880, 2157150, 2157150

Offset: 1

Peter Bala, Aug 02 2012

This is the table of f(n,k) in the notation of Carlitz (p.123). The triangle enumerates two-line arrays of positive integers
............a_1 a_2 ... a_n..........
............b_1 b_2 ... b_n..........
such that
1) max(a_i, b_i) <= min(a_(i+1), b_(i+1)) for 1 <= i <= n-1
2) max(a_i, b_i) <= i for 1 <= i <= n
3) a_n = b_n = k.
See A071948 and A193091 for other two-line array enumerations.
It appears that the row reverse array is the Riordan array (f(x), g(x)), where f(x) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + 818*x^5 + ... is the g.f. of A003168 and g(x) = x + 3*x^2 + 14*x^3 + 79*x^4 + 494*x^5 + 3294*x^6 + ... is the g.f. of A003169. - Peter Bala, Nov 26 2024

			Triangle begins
.n\k.|..1....2....3....4....5....6
= = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..1....1
..3..|..1....4....4
..4..|..1....7...21...21
..5..|..1...10...47..126..126
..6..|..1...13...82..324..818..818
...
T(4,2) = 7: The 7 two-line arrays are
...1 1 1 2....1 1 2 2....1 2 2 2....1 1 1 2
...1 1 1 2....1 1 2 2....1 2 2 2....1 1 2 2
...........................................
...1 1 2 2....1 1 2 2....1 2 2 2...........
...1 1 1 2....1 2 2 2....1 1 2 2...........

L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.

Cf. A003168 (main diagonal), A211789 (row sums).

Cf. A003169, A071948, A193091.

Recurrence equation:
T(1,1) = 1; T(n,n) = T(n,n-1); T(n+1,k) = Sum_{j = 1..k} (2*k-2*j+1)*T(n,j) for 1 <= k <= n.
T(n+1,k+1) = (1/n) * ((n - k)*Sum_{i = 0..k} C(n, k-i)*C(2*n+i, i) + Sum_{i = 1..k} C(n, k-i)*C(2*n+i, i-1)).
Row reverse has production matrix
  1 1
  3 3 1
  5 5 3 1
  7 7 5 3 1
  ...
Main diagonal T(n,n) = A003168(n). Row sums A211789.

A371581 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) / (1 - xA(x)) )^2.

Original entry on oeis.org

1, 2, 13, 108, 1018, 10352, 110724, 1227752, 13986369, 162708728, 1924866345, 23085868814, 280060995369, 3430479393210, 42369377446083, 527064922683286, 6597825455023465, 83050276697808472, 1050551595788997356, 13347641275527720048, 170259412138463630535

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Cf. A006013, A211789, A365146, A371582.

Cf. A371574, A365192.

a(n, r=2, s=1, t=5, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A365192.

A378611 a(n) = Sum_{k=0..n} binomial(2n+k-1,k) binomial(n-1,n-k).

Original entry on oeis.org

1, 2, 14, 104, 806, 6412, 51908, 425476, 3520070, 29332940, 245841284, 2070093632, 17499188924, 148414157816, 1262280506144, 10762045739644, 91951462167110, 787113739061260, 6749009521216052, 57954807274992208, 498334047795436276, 4290199618047230824

Offset: 0

Seiichi Manyama, Dec 01 2024

nonn

Cf. A002002, A378612, A378613.

Cf. A211789, A259554.

a(n) = sum(k=0, n, binomial(2*n+k-1, k)*binomial(n-1, n-k));

a(n) = [x^n] 1/(1 - x/(1 - x))^(2*n).
a(n) = (1/2)^n * [x^(2*n)] 2/(1 - x/(1 - x))^n for n > 0.
a(n) = 2 * A259554(n) for n > 0.

A378668 G.f. A(x) satisfies A(x) = 1/( 1 - xA(x)^2/(1 - xA(x)^2) )^2.

Original entry on oeis.org

1, 2, 13, 112, 1104, 11778, 132374, 1543740, 18505996, 226632616, 2823110349, 35659080952, 455652487060, 5879489288828, 76502741016012, 1002670573618324, 13224761472453756, 175404372357915096, 2338003752387818372, 31302169754776944512, 420760252068869028028

Offset: 0

Seiichi Manyama, Dec 02 2024

nonn

Cf. A243667, A378610, A378669.
Cf. A010683, A211789, A378670.
Cf. A378613.

a(n) = 2*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(4*n+k+2, n)/(4*n+k+2));

a(n, r=2, s=1, t=5, u=4) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f.: exp( 1/2 * Sum_{k>=1} A378613(k) * x^k/k ).
G.f.: B(x)^2 where B(x) is the g.f. of A243667.
a(n) = 2 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(4*n+k+2,n)/(4*n+k+2).
a(n) = 2 * Sum_{k=0..n} binomial(4*n+k+2,k) * binomial(n-1,n-k)/(4*n+k+2).
G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(5/2)/(1 - x*A(x)^2) )^2.

A371582 G.f. satisfies A(x) = ( 1 + xA(x)^3 / (1 - xA(x)) )^2.

Original entry on oeis.org

1, 2, 15, 146, 1623, 19526, 247516, 3256118, 44037023, 608484766, 8552832116, 121908218724, 1757915510695, 25598937436696, 375916184707142, 5560517754432772, 82774606577536376, 1239110145377709862, 18641533742708676711, 281697878640036748684

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Cf. A006013, A211789, A365146, A371581.

Cf. A371575.

a(n, r=2, s=1, t=6, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

cp's OEIS Frontend

A369012 Expansion of (1/x) * Series_Reversion( x * (1-x/(1-x))^3 ).

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A286785 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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A369011 Expansion of (1/x) * Series_Reversion( x * (1-x^3/(1-x))^2 ).

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A368957 Expansion of (1/x) * Series_Reversion( x * (1-x^2/(1-x))^2 ).

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A378610 Expansion of (1/x) * Series_Reversion( x * (1 - x/(1 - x))^4 ).

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A211788 Triangle enumerating certain two-line arrays of positive integers.

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A371581 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) / (1 - x*A(x)) )^2.

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A378611 a(n) = Sum_{k=0..n} binomial(2*n+k-1,k) * binomial(n-1,n-k).

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A378668 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^2/(1 - x*A(x)^2) )^2.

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A371581 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) / (1 - xA(x)) )^2.

A378611 a(n) = Sum_{k=0..n} binomial(2n+k-1,k) binomial(n-1,n-k).

A378668 G.f. A(x) satisfies A(x) = 1/( 1 - xA(x)^2/(1 - xA(x)^2) )^2.

A371582 G.f. satisfies A(x) = ( 1 + xA(x)^3 / (1 - xA(x)) )^2.