Dmitry Kruchinin - cp's OEIS frontend

User: Dmitry Kruchinin

Dmitry Kruchinin has authored 8 sequences.

A226952 Triangle of coefficients of Faber polynomials for (3x - sqrt(x^2 - 4x))/2.

0, -1, 1, -1, -2, 1, -4, 0, -3, 1, -13, -4, 2, -4, 1, -46, -10, -5, 5, -5, 1, -166, -36, -6, -8, 9, -6, 1, -610, -126, -28, 0, -14, 14, -7, 1, -2269, -456, -92, -24, 10, -24, 20, -8, 1, -8518, -1674, -333, -63, -27, 27, -39, 27, -9, 1

Offset: 0

Dmitry Kruchinin, Jun 24 2013

			Triangle begins as:
    0;
   -1,   1;
   -1,  -2,  1;
   -4,   0, -3,  1;
  -13,  -4,  2, -4,  1;
  -46, -10, -5,  5, -5,  1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

[[n eq 0 and k eq 0 select 0 else k eq n select 1 else n*(&+[ (-1)^j*j*Binomial(j+k,k)*Binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)): j in [1..n-k]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 29 2019

T[n_,k_]:= If[n==k==0, 0, If[k==n, 1, n*Sum[(-1)^j*j*Binomial[j+k, k]* Binomial[2*n-2*k-j-1, n-k-1]/((j+k)*(n-k)), {j, 1, n-k}]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 29 2019 *)

T(n,k):=if n=0 and k=0 then 0 else if n=k then 1 else n*sum(binomial(i+k,k)*(i)*binomial(2*(n-k)-i-1,n-k-1)*(-1)^(i)/((i+k)*(n-k)),i,1,n-k);

{T(n,k) = if(n==0 && k==0, 0, if(k==n, 1, n*sum(j=1,n-k, (-1)^j*j* binomial(j+k, k)*binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)))))}; \\ G. C. Greubel, Apr 29 2019

def T(n, k):
  if (k==n==0): return 0
  elif (k==n): return 1
  else: return n*sum((-1)^j*j* binomial(j+k, k)*binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)) for j in (1..n-k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 29 2019

G.f.: log(1 + (1 - sqrt(1-4*t))/2 - t*x) = Sum_{n>0} Sum_{k=0..n} T(n,k) * x^k * t^n/n.

T(n,k) = n*Sum_{j=1..n-k} binomial(j+k,k)*(j)*binomial(2*(n-k)-j-1, n-k-1)*(-1)^j/((j+k)*(n-k)), k

(-1)^(n+1) * Sum_{k=0..n} T(n,k) = 2*A181933(n).

T(n,0) = -A026641(n-1), n>0.

A220288 G.f. A(x) satisfies A(A(A(x))) = x+3x^2+9x^3.

Original entry on oeis.org

1, 1, 1, -8, 28, -26, -386, 2701, -5399, -42155, 358615, -354212, -10419524, 52825312, 236952352, -3103798967, -3013742105, 176201013745, -164790760103, -11763898514324, 27830312919316, 992172068848126, -3681957974446718, -103284064687144985, 528045230825074855

Offset: 1

Dmitry Kruchinin, Dec 09 2012

sign

Seiichi Manyama, Table of n, a(n) for n = 1..200

C. A141118, A220110.

t[n_, m_] := t[n, m] = 1/3*(3^(n - m)* Sum[Binomial[j, n - 3*m + 2*j]*Binomial[m, j], {j, 0, m}] - Sum[t[k, m]*Sum[t[n, i]*t[i, k], {i, k, n}], {k, m + 1, n - 1}] - Sum[t[n, i]*t[i, m], {i, m + 1, n - 1}]); t[n_, n_] = 1; Table[t[n, 1], {n, 1, 25}] (* Jean-François Alcover, Feb 22 2013 *)

T(n, m):=if n=m then 1 else 1/3*(3^(n-m)*sum(binomial(j,n-3*m+2*j)*binomial(m,j),j,0,m)-sum(T(k, m)*sum(T(n, i)*T(i, k), i, k, n), k, m+1, n-1)-sum(T(n, i)*T(i, m), i, m+1, n-1));
makelist(T(n,1),n,1,7);

a(n)=T(n,1), T(n, m)=1/3*(3^(n-m)*sum(j=0..m, binomial(j,n-3*m+2*j)*binomial(m,j))-sum(k=m+1..n-1, T(k, m)*sum(, i=k..n, T(n, i)*T(i, k)))-sum(i=m+1..n-1, T(n, i)*T(i, m))), T(n,n)=1.

A220112 E.g.f. A(x) satisfies A(A(x)) = (1/4)log(1/(1-4x)).

Original entry on oeis.org

1, 2, 10, 80, 872, 11928, 195072, 3702080, 80065792, 1950808000, 53016791360, 1587229842688, 51619520360960, 1808576831681536, 68562454975587328, 2830905156661645312, 124395772159835529216, 5504660984739184156672, 250011277837808237105152, 14799530615476409472303104

Offset: 1

Dmitry Kruchinin, Dec 05 2012

sign

a(23) = -4050933314339181211663673622528 is the first negative term. - Vladimir Reshetnikov, Aug 15 2021

Comtet, L; Advanced Combinatorics (1974 edition), D. Reidel Publishing Company, Dordrecht - Holland, pp. 147-148.

Vladimir Reshetnikov, Table of n, a(n) for n = 1..281
Gottfried Helms, Coefficients for fractional iterates exp(x)-1
Dmitry Kruchinin and Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986 [math.CO], 2013

Cf. A052122, A052123, A184011.

A := proc(n, m) option remember; if n = m then 1 else
1/2*(4^(n-m)*(-1)^(n-m)*Stirling1(n,m) - add(A(n,k)*A(k,m), k =m+1..n-1)) fi end: a := n -> A(n,1): seq(a(n), n = 1..23); # Peter Luschny, Aug 15 2021

t[n_, m_] := t[n, m] = 1/2*(4^(n - m)*(-1)^(n - m)*StirlingS1[n, m] - Sum[t[n, i]*t[i, m], {i, m+1, n-1}]); t[n_, n_] = 1; Table[t[n, 1], {n, 1, 20}] (* Jean-François Alcover, Feb 22 2013 *)

T(n,m):=if n=m then 1 else 1/2*(4^(n-m)*(-1)^(n-m)*stirling1(n,m)-sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist((T(n,1)),n,1,10);

a(n) = T(n,1), T(n,m) = (1/2)*(4^(n-m)*(-1)^(n-m)*Stirling1(n,m) - Sum_{i=m+1..n-1} T(n,i)*T(i,m)), T(n,n)=1.

More terms from Vladimir Reshetnikov, Aug 15 2021

A220113 E.g.f. A(x)=sum{n>0, a(n)x^(2n-1)/(2n-1)!} satisfies A(A(x))=sin(2*x)/2.

Original entry on oeis.org

1, -2, -12, -424, -29808, -2966816, -237449920, 76118167936, 84317834342656, 53499781544238592, 20080969948883956736, -10740526073453596649472, -31099457241702481710116864

Offset: 1

Dmitry Kruchinin, Dec 05 2012

sign

Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986

Cf. A048602.

t[n_, m_] := t[n, m] = 1/2*(2^(n - 2*m)*(((-1)^(n-m) + 1)* Sum[(2*i - m)^n*Binomial[m, i]*(-1)^((n+m)/2 - i), {i, 0, m/2}])/m! - Sum[t[n, i]*t[i, m], {i, m+1, n-1}]); t[n_, n_] = 1; Table[ t[2*n-1, 1], {n, 1, 13}] (* Jean-François Alcover, Feb 22 2013 *)

T(n,m):=if n=m then 1 else 1/2*(2^(n-2*m)*(((-1)^(n-m)+1)*sum((2*i-m)^n*binomial(m,i)*(-1)^((n+m)/2-i),i,0,m/2))/m!-sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist(((T3(2*n-1,1))),n,1,7);

a(n)=T(2*n-1,1), T(n,m)=1/2*(2^(n-2*m)*(((-1)^(n-m)+1)*sum(i=0..m/2, (2*i-m)^n*binomial(m,i)*(-1)^((n+m)/2-i)))/m!-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1.

A220110 Expansion of A(x) satisfying A(A(x)) = x+2x^2+4x^3.

Original entry on oeis.org

1, 1, 1, -3, 5, 1, -39, 117, 13, -1311, 3441, 9525, -78603, 16961, 1520521, -3649323, -28760163, 144787265, 601582689, -5374096875, -15170850555, 225456060897, 461284881657, -11141961064971, -15963771799251, 647040052660257, 569313149887057

Offset: 1

Dmitry Kruchinin, Dec 05 2012

sign

			First column of
1;
1,1;
1,2,1;
-3,3,3,1;
5,-4,6,4,1;
1,5,-2,10,5,1;
-39,6,3,4,15,6,1;
117,-57,9,3,15,21,7,1;
13,128,-56,8,10,32,28,8,1;
-1311,201,84,-44,6,30,56,36,9,1;

Seiichi Manyama, Table of n, a(n) for n = 1..552
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986

t[n_, m_] := t[n, m] = 1/2*(2^(n-m)*Sum[Binomial[j, n - 3*m + 2*j]*Binomial[m, j], {j, 0, m}] - Sum[t[n, k]*t[k, m], {k, m+1, n-1}]); t[n_, n_] = 1; Table[t[n, 1], {n, 1, 27}] (* Jean-François Alcover, Feb 22 2013 *)

T(n,m):=if n=m then 1 else 1/2*(2^(n-m)*sum(binomial(j,n-3*m+2*j)*binomial(m,j),j,0,m)-sum(T(n,k)*T(k,m),k,m+1,n-1));
makelist((T(n,1)),n,1,10);

a(n)=T(n,1), 2*T(n,m)= 2^(n-m) *sum_{j=0..m} binomial(j,n-3*m+2*j) *binomial(m,j) -sum_{k=m+1..n-1} T(n,k)*T(k,m), n>m, T(n,n)=1.

A200144 The number of multinomial coefficients, based on a set of partitions of n into m positions, divisible by m entirely.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 64, 100, 121, 167, 213, 296, 354, 489, 594, 776, 964, 1254, 1511, 1951, 2378, 2986, 3643, 4564, 5483, 6841, 8245, 10099, 12190, 14862, 17783, 21636, 25849, 31184

Offset: 1

Dmitry Kruchinin, Nov 11 2011

nonn

If n is prime, then the number of multinomial coefficients, based on a set of partitions of n at position m, divided by m entirely, less 1 than the number of partitions of numbers for all m.

			n=7;
  Set of partitions of n into m=4 parts
[1,1,1,4]
[1,1,2,3]
[1,2,2,2]
number of different parts
[3,1]
[2,1,1]
[1,3]
Multinomial coefficient,  divisible by m
4!/(4*(1!*3!))=1
4!/(4*(2!*1!*1!))=2
4!/(4*(1!*3!))=1
Set of partitions of n into m=7 parts
[1,1,1,1,1,1,1]
number of different parts
[7]
Multinomial coefficient,  divisible by m
7!/(7*(7!))=1/7

Dmitry Kruchinin The number of multinomial coefficients based on a set of partitions of n into k parts and divided by k evenly

/* count number of partitions of n into m parts */
b(n, m):=if n

A190667 Expansion of (1+2x)/(1-x^4-2x^3-2*x^2-x).

Original entry on oeis.org

1, 3, 5, 13, 30, 69, 160, 371, 859, 1990, 4610, 10679, 24738, 57306, 132750, 307517, 712367, 1650207, 3822725, 8855390, 20513621, 47520058, 110080805, 255003553, 590718900, 1368407674, 3169933385, 7343190086, 17010591104, 39405245720

Offset: 0

Dmitry Kruchinin, May 16 2011

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,2,1).

[ (n+1)*&+[ (Binomial(k, n-k+1)*Fibonacci(k))/k: k in [1..n+1] ]: n in [0..35] ]; // Klaus Brockhaus, May 17 2011

[ n eq 1 select 1 else n eq 2 select 3 else n eq 3 select 5 else n eq 4 select 13 else Self(n-1)+2*Self(n-2)+2*Self(n-3)+Self(n-4): n in [1..36] ]; // Klaus Brockhaus, Jun 01 2011

CoefficientList[Series[(1+2x)/(1-x^4-2x^3-2x^2-x),{x,0,40}],x] (* or *) LinearRecurrence[ {1,2,2,1},{1,3,5,13},40] (* Harvey P. Dale, Feb 25 2023 *)

a(n):=(n+1)*sum(binomial(k,n-k+1)*fib(k)/k,k,1,n+1);makelist(a(n),n,0,35);

a(n):=if n<0 then 0 else if n=0 then 1 else if n=1 then 3 else if n=2 then 5 else if n=3 then 13 else a(n-1)+2*a(n-2)+2*a(n-3)+a(n-4);

a(n) = (n+1)*sum(k=1..n+1, binomial(k, n-k+1)*A000045(k)/k).

a(n) = a(n-1)+2*a(n-2)+2*a(n-3)+a(n-4), a(0)=1, a(1)=3, a(2)=5, a(3)=13.

A190733 Expansion of (4x+2)/(1+sqrt(1-4x-4*x^2)).

Original entry on oeis.org

1, 3, 5, 15, 49, 175, 657, 2559, 10241, 41855, 173953, 732927, 3123457, 13439743, 58307841, 254779391, 1120247809, 4952864767, 22005184513, 98196398079, 439923990529, 1977917169663, 8921667641345, 40361657696255, 183092192411649, 832634240106495, 3795237359190017

Offset: 0

Dmitry Kruchinin, May 26 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

CoefficientList[Series[(4x+2)/(1+Sqrt[1-4x-4x^2]),{x,0,40}],x] (* Harvey P. Dale, Mar 20 2015 *)

a(n):=(n+1)*sum(binomial(k,n-k+1)/k*binomial(2*k-2,k-1)/k,k,1,(n+1))

x='x+O('x^66); /* that many terms */
Vec((4*x+2)/(1+sqrt(1-4*x-4*x^2))) /* show terms */ /* Joerg Arndt, May 27 2011 */

a(n) = (n+1)*sum(k=1..n+1,binomial(k,n-k+1)*Catalan(k-1)/k).

D-finite with recurrence: (n+1)*a(n) +(-n+1)*a(n-1) +14*(-n+2)*a(n-2) +20*(-n+3)*a(n-3) +8*(-n+4)*a(n-4)=0. - R. J. Mathar, Jan 25 2020

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User: Dmitry Kruchinin

A226952 Triangle of coefficients of Faber polynomials for (3*x - sqrt(x^2 - 4*x))/2.

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A220288 G.f. A(x) satisfies A(A(A(x))) = x+3*x^2+9*x^3.

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A220112 E.g.f. A(x) satisfies A(A(x)) = (1/4)*log(1/(1-4*x)).

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A220113 E.g.f. A(x)=sum{n>0, a(n)x^(2*n-1)/(2*n-1)!} satisfies A(A(x))=sin(2*x)/2.

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A220110 Expansion of A(x) satisfying A(A(x)) = x+2x^2+4x^3.

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A200144 The number of multinomial coefficients, based on a set of partitions of n into m positions, divisible by m entirely.

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A190667 Expansion of (1+2*x)/(1-x^4-2*x^3-2*x^2-x).

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A226952 Triangle of coefficients of Faber polynomials for (3x - sqrt(x^2 - 4x))/2.

A220288 G.f. A(x) satisfies A(A(A(x))) = x+3x^2+9x^3.

A220112 E.g.f. A(x) satisfies A(A(x)) = (1/4)log(1/(1-4x)).

A220113 E.g.f. A(x)=sum{n>0, a(n)x^(2n-1)/(2n-1)!} satisfies A(A(x))=sin(2*x)/2.

A190667 Expansion of (1+2x)/(1-x^4-2x^3-2*x^2-x).

A190733 Expansion of (4x+2)/(1+sqrt(1-4x-4*x^2)).