Vladimir Kruchinin - cp's OEIS frontend

A382225 Triangle read by rows: T(n,k) = Sum_{i=k..n} C(i-1,i-k)*C(i,k).

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 25, 13, 1, 1, 15, 65, 73, 21, 1, 1, 21, 140, 273, 171, 31, 1, 1, 28, 266, 798, 871, 346, 43, 1, 1, 36, 462, 1974, 3321, 2306, 631, 57, 1, 1, 45, 750, 4326, 10377, 11126, 5335, 1065, 73, 1, 1, 55, 1155, 8646, 28017, 42878, 31795, 11145, 1693, 91, 1

Offset: 0

Vladimir Kruchinin, Mar 19 2025

Triangle T(n,k) of minors of the main diagonal of Pascal's matrix, n -size matrix, k - number of element of diagonal.

			Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   7,   1;
  1, 10,  25,  13,   1;
  1, 15,  65,  73,  21,  1;
  1, 21, 140, 273, 171, 31, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Wikipedia, Minor (linear algebra)

Columns k=0-3 give: A000012, A000217, A001296(n-1) for n>=1, A107963(n-3) for n>=3.
Row sums give A024718.
T(n+1,n) gives A002061(n+1).
Cf. A007318, A184173.

T:= proc(n, k) option remember; `if`(n<0, 0,
      T(n-1, k)+binomial(n-1, k-1)*binomial(n, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 20 2025

A382225[n_, k_] := A382225[n, k] = If[k == n, 1, A382225[n-1, k] + Binomial[n-1, k-1]*Binomial[n, k]];
Table[A382225[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 22 2025 *)

h[i,j]:=binomial(i+j-3,i-1);
for n:1 thru 7 do
    if n=1 then print([1])
    else (M:genmatrix(h,n,n),
          print(makelist(determinant(minor(M,k,k)),k,1,n))
         );

G.f.: 1/(1-x) * ((1-x*(1-y))/(2*(sqrt((1-x*(1+y))^2-4*x^2*y)))+1/2).

T(n,k) = T(n-1,k)+C(n-1,k-1)*C(n,k).

A380048 a(n) = A000045(n) * A001003(n).

Original entry on oeis.org

0, 1, 3, 22, 135, 985, 7224, 55627, 436653, 3503666, 28537245, 235558347, 1965437136, 16552173909, 140505456663, 1200968926590, 10327551834411, 89286693775373, 775611959272392, 6766372185052247, 59256729852910425, 520754062920338026, 4590973472772299193, 40591542233796808247

Offset: 0

Vladimir Kruchinin, Jan 11 2025

Cf. A000045, A001003.

G.f.: 1/(4*x)-sqrt(16*x^2*(3+6*x+2*x^2)+32*x^2*sqrt(1-6*x-33*x^2+6*x^3+x^4))/(16*sqrt(5)*x^2).

a(n) = A000045(n) * A001003(n).

A373614 a(n) = Fibonacci(n)^2 * Catalan(n).

Original entry on oeis.org

0, 1, 2, 20, 126, 1050, 8448, 72501, 630630, 5620472, 50807900, 465643906, 4313336832, 40331298100, 380115482760, 3607451824500, 34444346026230, 330647239219110, 3189220347667200, 30893105448487590, 300408447948394500, 2931423727834870320, 28696206742447216440, 281728667746183208850, 2773282854528632549376

Offset: 0

Vladimir Kruchinin, Jun 10 2024

nonn

Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Generating Functions of Products of Generalized Fibonacci numbers, Catalan and Harmonic Numbers, arXiv:2406.02937 [math.CO], 2024.

Cf. A000045, A000108, A007598, A119694.

gf := (2 * sqrt(-sqrt(16*x^2 - 12*x+1) - 6*x + 1) / (5*sqrt(10)*x) + 3*(1 -sqrt((-2*sqrt(16*x^2 - 12*x + 1) - 42*x + 7) / 5 + 6*x))/(10*x)) + (1 -sqrt(4*x + 1)) / (5*x): assume(x > 0): ser := series(gf, x, 30):
seq(coeff(ser, x, n), n = 0..24); # Peter Luschny, Jun 11 2024

CoefficientList[Series[(2*Sqrt[-Sqrt[16*x^2 - 12*x + 1] - 6*x + 1]/(5*Sqrt[10]*x) + 3*(1 -Sqrt[(-2*Sqrt[16*x^2 - 12*x + 1] - 42*x + 7)/5 + 6*x])/(10*x)) + (1 -Sqrt[4*x + 1])/(5*x),{x,0,24},Assumptions->(x>0)],x] (* Stefano Spezia, Jun 11 2024 *)
(* A variant that does not need assumptions: *)
gf := ((2 Sqrt[1 - 2 x (Sqrt[5] + 3)] + Sqrt[2] (Sqrt[5] + 2) Sqrt[3 + Sqrt[5] - 8 x] + (Sqrt[5] + 3) (2 Sqrt[4 x + 1] - 5)) (Sqrt[5] - 3)) / (40 x);
Round[CoefficientList[Series[gf, {x, 0, 24}], x]]  (* Peter Luschny, Jun 11 2024 *)

G.f.: (2*sqrt(-sqrt(16*x^2 - 12*x + 1) - 6*x + 1)/(5*sqrt(10)*x) + 3*(1 -sqrt((-2*sqrt(16*x^2 - 12*x + 1) - 42*x + 7)/5 + 6*x))/(10*x)) + (1 -sqrt(4*x + 1))/(5*x).
a(n) = A007598(n)*A000108(n).
D-finite with recurrence n*(n-1)*(n+1)*a(n) -4*n*(2*n-1)*(n-1)*a(n-1) -8*(n-1)*(2*n-1)*(2*n-3)*a(n-2) +8*(2*n-5)*(2*n-1)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jun 12 2024

A373622 a(n) = A000032(n)A000045(n)A000108(n).

Original entry on oeis.org

0, 1, 6, 40, 294, 2310, 19008, 161733, 1411410, 12563408, 113624940, 1041158846, 9645100416, 90182859700, 849966450840, 8066498833800, 77019930780030, 739349587508730, 7131313919822400, 69079082238199110, 671733716498945100, 6554862704411317920, 64166669054324268120, 629964451984076275950

Offset: 0

Vladimir Kruchinin, Jun 11 2024

nonn

Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Generating Functions of Products of Generalized Fibonacci numbers, Catalan and Harmonic Numbers, arXiv:2406.02937 [math.CO], 2024.

Cf. A000032, A000045, A000108, A119694.

gf := (sqrt(-10*sqrt(16*x^2 - 12*x + 1) - 60*x + 35) - 5) / (10*x):
ser := series(gf, x, 32): seq(coeff(ser, x, n), n = 0..22);
# Peter Luschny, Jun 11 2024

CoefficientList[Series[(Sqrt[(-2*Sqrt[16*x^2-12*x+1]-42*x+7)/5+6*x]-1)/(2*x),{x,0,23}],x] (* Stefano Spezia, Jun 11 2024 *)

G.f.: (sqrt((-2*sqrt(16*x^2 - 12*x + 1) - 42*x + 7)/5 + 6*x) - 1)/(2*x).

D-finite with recurrence n*(n+1)*a(n) -6*n*(2*n-1)*a(n-1) +4*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jun 12 2024

A372252 a(n) = A000045(n+1)*A000292(n) (product of a Fibonacci number and a tetrahedral number).

Original entry on oeis.org

0, 1, 8, 30, 100, 280, 728, 1764, 4080, 9075, 19580, 41184, 84812, 171535, 341600, 671160, 1303152, 2503896, 4766340, 8997450, 16856840, 31366181, 58001768, 106646400, 195065000, 355074525, 643465368, 1161281394

Offset: 0

Vladimir Kruchinin, Apr 24 2024

nonn

Cf. A000045, A000292.

G.f.: -x*((x^2+1)*(x^2-4*x-1))/(x^2+x-1)^4.

A372199 a(n) = n! * F(n) * H(n), where F(n) is the n-th Fibonacci number and H(n) the n-th harmonic number.

Original entry on oeis.org

1, 3, 22, 150, 1370, 14112, 169884, 2301264, 34903584, 584575200, 10728401760, 214047774720, 4614042856320, 106866549054720, 2646889430976000, 69814736722483200, 1953778728154982400, 57822137143219814400, 1804373878844546150400, 59213693468692224000000

Offset: 1

Vladimir Kruchinin, Apr 21 2024

nonn

Cf. A000045, A001008, A002805.

H := proc(n)
    add(1/i,i=1..n) ;
end proc:
A372199 := proc(n)
    n!*A000045(n)*H(n) ;
end proc:
seq(A372199(n),n=1..70) ; # R. J. Mathar, Apr 24 2024

a[n_] := n! Fibonacci[n] HarmonicNumber[n]; Array[a,20] (* Stefano Spezia, Apr 22 2024 *)

E.g.f.: (5*x*log(-x^2 - x + 1) - sqrt(5)*(x - 2)*(log(2 - (sqrt(5) + 1)*x) -log((sqrt(5) - 1)*x + 2))) / (10*x*(x^2 + x - 1)).
a(n) = n! * A000045(n) * A001008(n) / A002805(n).
a(n) = A000045(n) * A000254(n) / A002805(n). - R. J. Mathar, Apr 24 2024
D-finite with recurrence 5*a(n) +5*(-2*n+1)*a(n-1) +(-5*n^2+10*n+1)*a(n-2) +(10*n^3-45*n^2+58*n-14)*a(n-3) +(5*n^4-40*n^3+109*n^2-108*n+16)*a(n-4) +2*(n-4)^3*a(n-5) +(n-4)^2*(n-5)^2*a(n-6)=0. - R. J. Mathar, Apr 24 2024

A372210 Product of n!, n-th Pell number and n-th harmonic number.

Original entry on oeis.org

1, 6, 55, 600, 7946, 123480, 2208492, 44710272, 1011177360, 25274905920, 692042185440, 20602098316800, 662620120237440, 22898921925035520, 846245264387040000, 33303963647943475200, 1390631677349880268800, 61407154400075559936000, 2859166138267857522585600

Offset: 1

Vladimir Kruchinin, Apr 22 2024

nonn

Cf. A000129, A000142, A000254, A001008, A002805, A052631.

E.g.f.: (2*x*log(-x^2-2*x+1)+(sqrt(2)-sqrt(2)*x)*log(-((sqrt(2)+1)*x-1) / ((sqrt(2)-1)*x+1)))/(4*(x^2+2*x-1)).
a(n) = n!*A000129(n)*A001008(n)/A002805(n).
D-finite with recurrence 8*a(n) +16*(-2*n+1)*a(n-1) +(16*n^2-32*n+25)*a(n-2) +4*(8*n^3-36*n^2+47*n-13)*a(n-3) +2*(2*n-5)*(2*n^3-11*n^2+17*n-7)*a(n-4) +4*(n-4)^3*a(n-5) +(n-4)^2*(n-5)^2*a(n-6)=0. - R. J. Mathar, Apr 24 2024

A372216 Product of n-th Pell number and n-th Catalan number: a(n) = A000129(n)*A000108(n).

Original entry on oeis.org

1, 4, 25, 168, 1218, 9240, 72501, 583440, 4789070, 39940888, 337490426, 2883046320, 24858176900, 216046612080, 1890737146125, 16647522481440, 147365806700310, 1310740896927000, 11708311265437710, 104989351441019760

Offset: 1

Vladimir Kruchinin, Apr 22 2024

nonn

Cf. A000108, A000129.

CoefficientList[Series[1/(2*x)-Sqrt[Sqrt[-16*x^2-8*x+1]+4*x+3]/(4*x),{x,0,20}],x] (* Stefano Spezia, Apr 22 2024 *)

G.f.: 1/(2*x)-sqrt(sqrt(-16*x^2-8*x+1)+4*x+3)/(4*x).

D-finite with recurrence +n*(n+1)*a(n) -4*n*(2*n-1)*a(n-1) -4*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Apr 24 2024

A372015 Product of Fibonacci and self-convolution of Fibonacci numbers: a(n) = A000045(n+1)*A001629(n+1).

Original entry on oeis.org

0, 1, 4, 15, 50, 160, 494, 1491, 4420, 12925, 37380, 107136, 304764, 861445, 2421700, 6775755, 18879734, 52413856, 145038890, 400183575, 1101277060, 3023462521, 8282790024, 22646131200, 61805595000, 168399404425, 458128878724, 1244567262471, 3376576740410, 9149594423200

Offset: 0

Vladimir Kruchinin, Apr 15 2024

Conjecture: a(n) is the total number of pairs of adjacent parts that are the same color in all n-color compositions of n+1. - John Tyler Rascoe, Jul 30 2024

John Tyler Rascoe, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (5,-5,-5,5,-1).

Cf. A000045, A001629, A088305.

a := proc(n) option remember; if n < 3 then return n^2 fi;
-((2 - 2*n^2 + n)*a(n - 1) + (1 - 2*n^2 + 3*n)*a(n - 2) + n^2*a(n - 3))/(n - 1)^2 end: seq(a(n), n = 0..29);  # Peter Luschny, Apr 16 2024

CoefficientList[Series[x(1-x)/((1+x)*(1-3*x+x^2)^2),{x,0,29}],x] (* Stefano Spezia, Apr 16 2024 *)

A_x(N)= {my(x='x+O('x^N)); concat([0],Vec(x*(1-x)/((1+x)*(1-3*x+x^2)^2)))}
A_x(40) \\ John Tyler Rascoe, Jul 29 2024

a(n) = F(n+1)*((n+2)*F(n) + (n)*F(n+2))/5 where F(n) = A000045(n) is the Fibonacci numbers.

G.f.: x*(1-x)/((1+x)*(1-3*x+x^2)^2).

A371986 Product of Lucas and Catalan numbers: a(n) = A000032(n)*A000108(n).

Original entry on oeis.org

2, 1, 6, 20, 98, 462, 2376, 12441, 67210, 369512, 2065908, 11698414, 66979864, 387050900, 2254552920, 13223768580, 78034377690, 462961545090, 2759796408600, 16522143563310, 99295449593340, 598836351581520, 3622983967834920, 21982916983078350, 133739841802846968

Offset: 0

Vladimir Kruchinin, Apr 15 2024

nonn

Cf. A000032, A000108, A098614, A119694, A127211, A216541.

From Peter Luschny, Apr 15 2024: (Start)
a := n -> ((2 - 2*sqrt(5))^n + (2 + 2*sqrt(5))^n) * GAMMA(n + 1/2) / (sqrt(Pi) * GAMMA(n + 2)): seq(simplify(a(n)), n = 0..24);
# With g.f.:
assume(x>0); f := sqrt(1 - 4*x*(4*x + 1)):
gf := (sqrt(1 + f - 2*x) + sqrt(5)*sqrt(1 - f - 2*x) - sqrt(2))/(sqrt(8)*x):
ser := series(gf, x, 26): seq(simplify(coeff(ser, x, n)), n = 0..24);
# Recurrence:
a := proc(n) option remember: if n < 2 then return [2, 1][n + 1] fi;
2*(2*n - 1)*(n*a(n - 1) + (4*n - 6)*a(n - 2)) / (n*(n + 1)) end:
seq(a(n), n=0..24);  (End)

a[n_] := a[n] = Switch[n, 0, 2, 1, 1, _, 2*(2n - 1)*(n*a[n - 1] + (4n - 6)*a[n - 2])/(n*(n + 1))];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jun 17 2024, after Peter Luschny *)

def A371986_gen(): # generator of terms
    a, b, n = 2, 1, 2
    while True:
        yield a
        a, b = b, (4*n - 2)*(n*b + (4*n - 6)*a) // (n*n + n)
        n += 1
def A371986_list(len):
    it =  A371986_gen()
    return [next(it) for _ in range(len)]
print(A371986_list(25))  # Peter Luschny, Apr 15 2024

G.f.: (5*sqrt(-sqrt(-16*x^2 - 4*x+1) - 2*x+1)) / (2*sqrt(10)*x) - (1 - sqrt(sqrt( -16*x^2 - 4*x+1) - 2*x + 1) / sqrt(2)) / (2*x).
E.g.f.: exp(x-sqrt(5)*x)*(BesselI(0, x-sqrt(5)*x) - BesselI(1, x-sqrt(5)*x) + exp(2*sqrt(5)*x) * (BesselI(0, x+sqrt(5)*x) - BesselI(1, x+sqrt(5)*x))). - Stefano Spezia, Apr 15 2024
From Peter Luschny, Apr 15 2024: (Start)
a(n) = 2*(2*n - 1)*(n*a(n - 1) + (4*n - 6)*a(n - 2)) / (n*(n + 1)) for n >= 2.
a(n) = ((2 - 2*sqrt(5))^n + (2 + 2*sqrt(5))^n) * Gamma(n + 1/2) / (sqrt(Pi) * Gamma(n + 2)).
a(n) ~ (2 + 2*sqrt(5))^n / (n*(n*Pi)^(1/2)). (End)

cp's OEIS Frontend

User: Vladimir Kruchinin

A382225 Triangle read by rows: T(n,k) = Sum_{i=k..n} C(i-1,i-k)*C(i,k).

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A380048 a(n) = A000045(n) * A001003(n).

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A373614 a(n) = Fibonacci(n)^2 * Catalan(n).

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A373622 a(n) = A000032(n)*A000045(n)*A000108(n).

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A372252 a(n) = A000045(n+1)*A000292(n) (product of a Fibonacci number and a tetrahedral number).

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A372199 a(n) = n! * F(n) * H(n), where F(n) is the n-th Fibonacci number and H(n) the n-th harmonic number.

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A372210 Product of n!, n-th Pell number and n-th harmonic number.

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A372216 Product of n-th Pell number and n-th Catalan number: a(n) = A000129(n)*A000108(n).

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A372015 Product of Fibonacci and self-convolution of Fibonacci numbers: a(n) = A000045(n+1)*A001629(n+1).

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A373622 a(n) = A000032(n)A000045(n)A000108(n).