A002294 -id:A002294 - cp's OEIS frontend

A264774 Triangle T(n,k) = binomial(5n - 4k, 4n - 3k), 0 <= k <= n.

1, 5, 1, 45, 6, 1, 455, 55, 7, 1, 4845, 560, 66, 8, 1, 53130, 5985, 680, 78, 9, 1, 593775, 65780, 7315, 816, 91, 10, 1, 6724520, 736281, 80730, 8855, 969, 105, 11, 1, 76904685, 8347680, 906192, 98280, 10626, 1140, 120, 12, 1, 886163135, 95548245, 10295472, 1107568, 118755, 12650, 1330, 136, 13, 1

Offset: 0

Peter Bala, Nov 30 2015

Riordan array (f(x),x*g(x)), where g(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + ... is the o.g.f. for A002294 and f(x) = g(x)/(5 - 4*g(x)) = 1 + 5*x + 45*x^2 + 455*x^3 + 4845*x^4 + ... is the o.g.f. for A001449.

More generally, if (R(n,k))n,k>=0 is a proper Riordan array and m is a nonnegative integer and a > b are integers then the array with (n,k)-th element R((m + 1)*n - a*k, m*n - b*k) is also a Riordan array (not necessarily proper). Here we take R as Pascal's triangle and m = a = 4 and b = 3. See A092392, A264772, A264773 and A113139 for further examples.

			Triangle begins
  n\k |       0      1     2    3   4   5   6   7
------+---------------------------------------------
   0  |       1
   1  |       5      1
   2  |      45      6     1
   3  |     455     55     7    1
   4  |    4845    560    66    8   1
   5  |   53130   5985   680   78   9   1
   6  |  593775  65780  7315  816  91  10   1
   7  | 6724520 736281 80730 8855 969 105  11  1
...

Peter Bala, A 4-parameter family of embedded Riordan arrays
E. Lebensztayn, On the asymptotic enumeration of accessible automata, Section 2, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No. 3, 2010, 75-80, Section 2.
R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, CreateSpace Independent Publishing Platform 2006, Section 5.6, ISBN-13: 978-1502925244.

Cf. A001449 (column 0), A079589(column 1). Cf. A002294, A007318, A092392 (C(2n-k,n)), A113139, A119301 (C(3n-k,n-k)), A264772, A264773.

/* As triangle */ [[Binomial(5*n-4*k, 4*n-3*k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 02 2015

A264774:= proc(n,k) binomial(5*n - 4*k, 4*n - 3*k); end proc:
seq(seq(A264774(n,k), k = 0..n), n = 0..10);

Table[Binomial[5 n - 4 k, 4 n - 3 k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)

T(n,k) = binomial(5*n - 4*k, n - k).

O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(5*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(4*n + 1)*binomial(5*n,n)*x^n.

A324958 Triangle of coefficients T(n,k) of y^n in Product_{k=0..n-2} (n + (2n + k)y + ny^2), as read by rows of terms k = 0..2n-2, for n >= 1.

Original entry on oeis.org

1, 2, 4, 2, 9, 39, 60, 39, 9, 64, 432, 1160, 1584, 1160, 432, 64, 625, 5750, 22275, 47380, 60460, 47380, 22275, 5750, 625, 7776, 90720, 461160, 1343160, 2479464, 3029040, 2479464, 1343160, 461160, 90720, 7776, 117649, 1663893, 10489969, 38937360, 94679711, 158760987, 188149822, 158760987, 94679711, 38937360, 10489969, 1663893, 117649, 2097152, 34865152, 262635520, 1187049472, 3593318400, 7701010688, 12043471488, 13957194496, 12043471488, 7701010688, 3593318400, 1187049472, 262635520, 34865152, 2097152

Offset: 1

Paul D. Hanna, Mar 20 2019

			E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k starts
A(x,y) = x + (2*y^2 + 4*y + 2)*x^2/2! + (9*y^4 + 39*y^3 + 60*y^2 + 39*y + 9)*x^3/3! + (64*y^6 + 432*y^5 + 1160*y^4 + 1584*y^3 + 1160*y^2 + 432*y + 64)*x^4/4! + (625*y^8 + 5750*y^7 + 22275*y^6 + 47380*y^5 + 60460*y^4 + 47380*y^3 + 22275*y^2 + 5750*y + 625)*x^5/5! + (7776*y^10 + 90720*y^9 + 461160*y^8 + 1343160*y^7 + 2479464*y^6 + 3029040*y^5 + 2479464*y^4 + 1343160*y^3 + 461160*y^2 + 90720*y + 7776)*x^6/6! + (117649*y^12 + 1663893*y^11 + 10489969*y^10 + 38937360*y^9 + 94679711*y^8 + 158760987*y^7 + 188149822*y^6 + 158760987*y^5 + 94679711*y^4 + 38937360*y^3 + 10489969*y^2 + 1663893*y + 117649)*x^7/7! + (2097152*y^14 + 34865152*y^13 + 262635520*y^12 + 1187049472*y^11 + 3593318400*y^10 + 7701010688*y^9 + 12043471488*y^8 + 13957194496*y^7 + 12043471488*y^6 + 7701010688*y^5 + 3593318400*y^4 + 1187049472*y^3 + 262635520*y^2 + 34865152*y + 2097152)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in e.g.f. A(x,y) begins:
1;
2, 4, 2;
9, 39, 60, 39, 9;
64, 432, 1160, 1584, 1160, 432, 64;
625, 5750, 22275, 47380, 60460, 47380, 22275, 5750, 625;
7776, 90720, 461160, 1343160, 2479464, 3029040, 2479464, 1343160, 461160, 90720, 7776;
117649, 1663893, 10489969, 38937360, 94679711, 158760987, 188149822, 158760987, 94679711, 38937360, 10489969, 1663893, 117649;
2097152, 34865152, 262635520, 1187049472, 3593318400, 7701010688, 12043471488, 13957194496, 12043471488, 7701010688, 3593318400, 1187049472, 262635520, 34865152, 2097152; ...

Cf. A324959, A002294.

Cf. A324960, A324305.

{T(n, k) = polcoeff(prod(m=0, n-2, n + (2*n+m)*y + n*y^2 +y*O(y^k)), k, y)}
for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))

{T(n,k) = my(A = serreverse( x*(1 - x*y +x*O(x^n) )^((1+y)^2/y)));
n!*polcoeff(polcoeff(A,n,x),k,y)}
for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))

E.g.f. A(x) = Sum_{n>=1} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k satisfies
(1) A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2).
(2) A(x,y) = Series_Reversion( x*(1 - x*y)^((1+y)^2/y) ), wrt x.
(3) A(x,y) = x/(1 - y*A(x))^((1+y)^2/y).
(4) A(x,y) = x*Sum_{n>=0} A(x,y)^n/n! * Product_{k=0..n-1} (1 + (k+2)*y + y^2).
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: A(x,y=0) = -LambertW(-x) = x*exp(-LambertW(-x)).
E.g.f. at y = 1: A(x,y=1) = x*G(x)^4, where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
FORMULAS INVOLVING TERMS.
Row sums: Sum_{k=0..2*n-2} T(n,k) = (5*n-2)!/(4*n-1)! for n >= 1.
T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
T(n,n-1) = A324959(n) for n >= 1.

A334610 a(n) is the total number of down-steps after the final up-step in all 4_1-Dyck paths of length 5n (n up-steps and 4n down-steps).

Original entry on oeis.org

0, 7, 58, 505, 4650, 44677, 443238, 4507461, 46744100, 492492330, 5257084420, 56734340091, 618001356458, 6785943435960, 75033214770640, 834733624099485, 9336542892778440, 104932793226255165, 1184421713336050590, 13421053387405062290, 152613573227667516580

Offset: 0

Andrei Asinowski, May 13 2020

nonn

A 4_1-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above y = -1, and ends at the x-axis.

			For n=1, a(1) = 7 is the total number of down-steps after the last up-step in Udddd, dUddd.

Stefano Spezia, Table of n, a(n) for n = 0..900
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002294, A334651, A334719, A334611, A334612.

a[n_] := 2 * Binomial[5*n + 7, n + 1]/(5*n + 7) - 4 * Binomial[5*n + 2, n]/(5*n + 2); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

a(n) = 2*binomial(5*(n+1)+2, n+1)/(5*(n+1)+2) - 4*binomial(5*n+2, n)/(5*n+2).

G.f.: ((1 - 2*x)*HypergeometricPFQ([2/5, 3/5, 4/5, 6/5], [3/4, 5/4, 3/2], 3125*x/256) - 1)/x. - Stefano Spezia, Apr 25 2023

A361240 Number of nonequivalent noncrossing triangular cacti with n triangles up to rotation and reflection.

Original entry on oeis.org

1, 1, 1, 4, 19, 124, 931, 7801, 68685, 630850, 5966610, 57808920, 571178751, 5737672339, 58455577800, 602859484608, 6283968796705, 66119472527814, 701526880303315, 7498841163925819, 80696081185766970, 873654670250482120, 9510760874015305314, 104056578392127906720

Offset: 0

Andrew Howroyd, Mar 06 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Column 3 of A361239.

Cf. A002294, A118970, A361237.

a(2*n) = (A361237(2*n) + A002294(n))/2; a(2*n+1) = (A361237(2*n+1) + A118970(n))/2.

A365668 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 2 * A(x)).

Original entry on oeis.org

0, 1, 7, 73, 905, 12354, 179305, 2715192, 42414021, 678476755, 11058588574, 182999237590, 3066447596459, 51926183715280, 887204891847960, 15276037569668880, 264797324173666845, 4617195655522976361, 80930337327794271445, 1425171253004955494215, 25202145191953299213490

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for 4-dimensional figurate numbers A001296 (with signs).

Eric Weisstein's World of Mathematics, Series Reversion

Cf. A001296, A002294, A064063, A365755, A366014, A366035, A366036, A366037.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 2 x)/(1 + x)^5, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 2^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(5*n,n-k-1) * 2^k for n > 0.

a(n) ~ sqrt(32 - 19*sqrt(5/2)) * 3^(4*n - 3/2) * 5^(3*n) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 3/2) * (25 + 34*sqrt(10))^n). - Vaclav Kotesovec, Sep 27 2023

A366035 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 3 * A(x)).

Original entry on oeis.org

0, 1, 8, 98, 1440, 23389, 404712, 7314724, 136476912, 2608808180, 50828498336, 1005682252458, 20152470321984, 408149824237302, 8341496306085040, 171812412714350280, 3562961488550366480, 74328284438252301996, 1558783863783469298016, 32844108784368485209320, 694957689921176181019520

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for 4-dimensional figurate numbers A002417 (with signs).

Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002294, A002417, A064087, A365668, A365755, A365816, A366015, A366036, A366037.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^5, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 3^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(5*n,n-k-1) * 3^k for n > 0.

a(n) ~ 2^(4*n - 1) * 5^(5*n + 1/2) / (sqrt(Pi) * n^(3/2) * 3^(7*n + 5/2)). - Vaclav Kotesovec, Sep 27 2023

A366036 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 4 * A(x)).

Original entry on oeis.org

0, 1, 9, 127, 2165, 40914, 824859, 17383720, 378373437, 8440227235, 191938302578, 4433259845898, 103716352560119, 2452629475989840, 58529969579982600, 1407775987050271920, 34092047564798908045, 830565580516900384329, 20342106952028722530603, 500573735323751221019425, 12370242700776737398052970

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for 4-dimensional figurate numbers A002418 (with signs).

Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002294, A002418, A064088, A365668, A365755, A365817, A366016, A366035, A366037.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 4 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 4 x)/(1 + x)^5, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 4^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(5*n,n-k-1) * 4^k for n > 0.

a(n) ~ sqrt(34*sqrt(6) - 81) * 2^(n - 11/4) * 3^(n - 5/4) * (3/2 - 1/sqrt(6))^(5*n) / (sqrt(Pi) * n^(3/2) * (3*sqrt(6) - 7)^n). - Vaclav Kotesovec, Sep 27 2023

A380553 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n(1-x)^(4n) ).

Original entry on oeis.org

1, 3, 25, 200, 1770, 16351, 158223, 1577328, 16112031, 167708890, 1772645419, 18974340640, 205263418940, 2240623110285, 24648785800540, 272994642782048, 3041495503591364, 34064252952038769, 383302465665133013, 4331178750570145160, 49126274119206904221, 559128033687856289017

Offset: 1

Paul D. Hanna, Feb 16 2025

nonn

Moebius transform of A118971.

			G.f.: A(x) = x + 3*x^2 + 25*x^3 + 200*x^4 + 1770*x^5 + 16351*x^6 + 158223*x^7 + 1577328*x^8 + 16112031*x^9 + 167708890*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 4*x^2 + 26*x^3 + 204*x^4 + 1771*x^5 + 16380*x^6 + 158224*x^7 + 1577532*x^8 + ... + A118971(n)*x^(n) + ...
which equals x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A346936, A034742, A380551, A380552, A118971, A002294, A008683.

\\ As the Moebius transform of A118971 \\
{a(n) = sumdiv(n,d, moebius(n/d) * binomial(5*d-1,d-1)*4/(5*d-1) )}
for(n=1,30,print1(a(n),", "))

\\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ) \\
{a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoef(x - sum(m=1,#V, subst(A,x, x^m*(1-x)^(4*m) +x*O(x^#V)) ),#V-1)); V[n+1]}
for(n=1,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
(2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(4*n) / (1 - x^n*(1-x)^(4*n)).
(3) x*F(x)^4 = Sum_{n>=1} a(n) * x^n/(1-x^n) where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.
(4) a(n) = Sum_{d|n} mu(n/d) * binomial(5*d-1,d-1)*4/(5*d-1), where mu is the Moebius function A008683.

A381986 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 17, 388, 14329, 727206, 46984729, 3689119624, 341097752657, 36302764864330, 4371463743828481, 587606216836328460, 87219196719691250185, 14168990447072685567214, 2500554381188629649979593, 476391652257266128440376336, 97447147561230881896398507553

Offset: 0

Seiichi Manyama, Mar 11 2025

nonn

Cf. A381984, A381985.

Cf. A001764, A002294, A382000.

a(n) = n!*sum(k=0, n, (2*k+1)^(n-k)*binomial(5*k+1, k)/((5*k+1)*(n-k)!));

Let F(x) be the e.g.f. of A382000. F(x) = B(x*A(x)^2) = exp( 1/3 * Sum_{k>=1} binomial(3*k,k) * (x*A(x)^2)^k/k ).

a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k) * A002294(k)/(n-k)!.

A384942 G.f. A(x) satisfies A(x) = 1 + x/A(-x*A(x))^5.

Original entry on oeis.org

1, 1, 5, -5, -135, -110, 3661, 16440, -1375, -827075, -8388505, 2298072, 496514205, 2782147265, 322830120, -164675585390, -1846591014842, -3084367863270, 84920580735040, 845318162940805, 4163798547024100, -18708392155753220, -503209620889452990, -3212928238924865090

Offset: 0

Seiichi Manyama, Jun 13 2025

sign

Column k=1 of A384945.
Cf. A002294, A213104.
Cf. A000012, A384894, A384896, A384941, A384943.

a(n, k=-1) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+2*j+k-1, j-1)*a(n-j, 5*j)/j));

See A384945.

cp's OEIS Frontend

A264774 Triangle T(n,k) = binomial(5*n - 4*k, 4*n - 3*k), 0 <= k <= n.

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A324958 Triangle of coefficients T(n,k) of y^n in Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2), as read by rows of terms k = 0..2*n-2, for n >= 1.

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A334610 a(n) is the total number of down-steps after the final up-step in all 4_1-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

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A361240 Number of nonequivalent noncrossing triangular cacti with n triangles up to rotation and reflection.

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A365668 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 2 * A(x)).

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A366035 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 3 * A(x)).

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A366036 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 4 * A(x)).

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A380553 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).

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A264774 Triangle T(n,k) = binomial(5n - 4k, 4n - 3k), 0 <= k <= n.

A324958 Triangle of coefficients T(n,k) of y^n in Product_{k=0..n-2} (n + (2n + k)y + ny^2), as read by rows of terms k = 0..2n-2, for n >= 1.

A334610 a(n) is the total number of down-steps after the final up-step in all 4_1-Dyck paths of length 5n (n up-steps and 4n down-steps).

A380553 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n(1-x)^(4n) ).

A381986 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)^2), where B(x) = 1 + xB(x)^3 is the g.f. of A001764.