A001517 -id:A001517 - cp's OEIS frontend

A168422 Number triangle with row sums given by quadruple factorial numbers A001813.

1, 1, 1, 7, 4, 1, 71, 39, 9, 1, 1001, 536, 126, 16, 1, 18089, 9545, 2270, 310, 25, 1, 398959, 208524, 49995, 7120, 645, 36, 1, 10391023, 5394991, 1301139, 190435, 18445, 1197, 49, 1, 312129649, 161260336, 39066076, 5828704, 589750, 41776, 2044, 64, 1

Offset: 0

Paul Barry, Nov 25 2009

Reversal of coefficient array for the polynomials P(n,x) = Sum_{k=0..n} (C(n+k,2k)*(2k)!/k!)*x^k*(1-x)^(n-k).

Note that P(n,x) = Sum_{k=0..n} A113025(n,k)*x^k*(1-x)^(n-k). Row sums are A001813.

			Triangle begins
          1
          1         1
          7         4        1
         71        39        9       1
       1001       536      126      16      1
      18089      9545     2270     310     25     1
     398959    208524    49995    7120    645    36    1
   10391023   5394991  1301139  190435  18445  1197   49  1
  312129649 161260336 39066076 5828704 589750 41776 2044 64 1
Production matrix begins
        1       1
        6       3       1
       40      20       5      1
      336     168      42      7     1
     3456    1728     432     72     9    1
    42240   21120    5280    880   110   11   1
   599040  299520   74880  12480  1560  156  13  1
  9676800 4838400 1209600 201600 25200 2520 210 15 1
Complete this with a top row (1,0,0,0,...) and invert: we get
    1
   -1   1
   -3  -3   1
   -5  -5  -5   1
   -7  -7  -7  -7   1
   -9  -9  -9  -9  -9   1
  -11 -11 -11 -11 -11 -11   1
  -13 -13 -13 -13 -13 -13 -13   1
  -15 -15 -15 -15 -15 -15 -15 -15   1
  -17 -17 -17 -17 -17 -17 -17 -17 -17   1

William P. Orrick, Table of n, a(n) for n = 0..10010

Column 1 is |A002119|.
Sum_{k=0..n} T(n,k) * 2^k, is A001517(n).
Cf. A079267.

T(n,k)={sum(j=k, n, (-1)^(j-k)*(2*n-j)!/((n-j)!*(j-k)!))/k!} \\ Andrew Howroyd, Mar 24 2023

def T(n,k):
    return(sum((-1)^(j-k) * binomial(2*n-j,n) * binomial(n,j)\
     * binomial(j,k) * factorial(n-j)\
     for j in range(k,n+1))) # William P. Orrick, Mar 24 2023

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

Corrected and extended by William P. Orrick, Mar 24 2023

A272648 a(n) = A002119(n) mod 7.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 6, 6, 0, 6, 0, 6, 6, 1, 1, 0, 1, 0, 1, 1, 6, 6, 0, 6, 0, 6, 6, 1, 1, 0, 1, 0, 1, 1, 6, 6, 0, 6, 0, 6, 6, 1, 1, 0, 1, 0, 1, 1, 6, 6, 0, 6, 0, 6, 6, 1, 1, 0, 1, 0, 1, 1, 6, 6, 0, 6, 0, 6, 6, 1, 1, 0, 1, 0, 1, 1, 6, 6, 0, 6

Offset: 0

N. J. A. Sloane, May 09 2016

Periodic with period 14.

Colin Barker, Table of n, a(n) for n = 0..1000
D. H. Lehmer, Arithmetical periodicities of Bessel functions, Annals of Mathematics, 33 (1932): 143-150. The sequence is on page 149.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A001517, A002119, A272647.

Cf. A010876.

b:=[1,-1];; for n in [3..95] do b[n]:=-2*(2*n-3)*b[n-1]+b[n-2]; od; a:=List(b,AbsInt) mod 7; # Muniru A Asiru, Sep 20 2018

f:=proc(n) option remember; if n = 0 then 1 elif n=1 then 1 else f(n-2)+(4*n-2)*f(n-1); fi; end;
[seq(f(n) mod 7, n=0..120)];

PadRight[{},120,{1,1,0,1,0,1,1,6,6,0,6,0,6,6}] (* Harvey P. Dale, Jun 07 2016 *)

Vec((1+x+x^3+x^5+x^6)*(1+6*x^7)/((1-x)*(1+x)*(1-x+x^2-x^3+x^4-x^5+x^6)*(1+x+x^2+x^3+x^4+x^5+x^6)) + O(x^50)) \\ Colin Barker, May 10 2016

G.f.: (1+x+x^3+x^5+x^6)*(1+6*x^7) / ((1-x)*(1+x)*(1-x+x^2-x^3+x^4-x^5+x^6)*(1+x+x^2+x^3+x^4+x^5+x^6)). - Colin Barker, May 10 2016

a(n) = (-m^6+18*m^5-122*m^4+384*m^3-549*m^2+270*m+24)*(7-5*(-1)^floor(n/7))/48, where m = (n mod 7). - Luce ETIENNE, Sep 21 2018

A307495 Expansion of Sum_{k>=0} k!((1 - sqrt(1 - 4x))/2)^k.

Original entry on oeis.org

1, 1, 3, 12, 57, 312, 1950, 13848, 111069, 998064, 9957186, 109305240, 1309637274, 17006109072, 237888664572, 3566114897520, 57030565449765, 969154436550240, 17439499379433690, 331268545604793240, 6624013560942038670, 139080391965533653200, 3059323407592802838180, 70355685298375014175440

Offset: 0

Ilya Gutkovskiy, Apr 10 2019

nonn

Catalan transform of A000142 (factorial numbers).
From Peter Bala, Jan 27 2020: (Start)
This sequence is the main diagonal of the lower triangular array formed by putting the sequence of factorial numbers in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.
    1
    1    1
    2    3    3
    6    9   12   12
   24   33   45   57   57
  120  153  198  255  312  312
  ...
Alternatively, the sequence can be obtained by multiplying the sequence of factorial numbers by the array A106566.
(End)

P. Bala, A note on the Catalan transform of a sequence

Cf. A000108, A000142, A013999, A100100, A106566, A307496.

nmax = 23; CoefficientList[Series[Sum[k! ((1 - Sqrt[1 - 4 x])/2)^k, {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1/(1 + ContinuedFractionK[-Floor[(k + 1)/2] (1 - Sqrt[1 - 4 x])/2, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k k!, {k, n}], {n, 23}]]

G.f.: 1 /(1 - x*c(x)/(1 - x*c(x)/(1 - 2*x*c(x)/(1 - 2*x*c(x)/(1 - 3*x*c(x)/(1 - 3*x*c(x)/(1 - ...))))))), a continued fraction, where c(x) = g.f. of Catalan numbers (A000108).
Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000142.
a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*k! for n > 0.
a(n) ~ exp(1) * n!. - Vaclav Kotesovec, Aug 10 2019

A369723 Expansion of e.g.f. exp( 2 * (1-sqrt(1-4*x)) ).

Original entry on oeis.org

1, 4, 24, 208, 2464, 37824, 720256, 16450816, 439245312, 13440572416, 464007387136, 17847329869824, 757011972726784, 35108108023349248, 1767517592731090944, 96007679735852498944, 5596725706163142393856, 348533116657888468402176

Offset: 0

Seiichi Manyama, Jan 30 2024

nonn

Cf. A001517, A101682, A243953, A369722, A369724.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*(1-sqrt(1-4*x)))))

a(0) = 1; a(n) = Sum_{k=0..n-1} 4^(n-k) * (n-1+k)! / (k! * (n-1-k)!).

a(n) = 2*(2*n-3)*a(n-1) + 16*a(n-2).

A369724 Expansion of e.g.f. exp( (5/2) * (1-sqrt(1-4*x)) ).

Original entry on oeis.org

1, 5, 35, 335, 4225, 67525, 1321075, 30751775, 832573025, 25745985125, 896177819875, 34698406783375, 1479737530398625, 68935386567921125, 3483762766656021875, 189846574063623209375, 11098195364856546690625, 692834276972696475053125

Offset: 0

Seiichi Manyama, Jan 30 2024

nonn

Cf. A001517, A101682, A243953, A369722, A369723.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(5/2*(1-sqrt(1-4*x)))))

a(0) = 1; a(n) = Sum_{k=0..n-1} 5^(n-k) * (n-1+k)! / (k! * (n-1-k)!).

a(n) = 2*(2*n-3)*a(n-1) + 25*a(n-2).

A079166 Square array read by antidiagonals of T(n,k)=(4k-2)*T(n,k-1)+T(n,k-2) with T(n,0)=1 and T(n,1)=n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 10, 7, 2, 1, 141, 71, 13, 3, 1, 2548, 1001, 132, 19, 4, 1, 56197, 18089, 1861, 193, 25, 5, 1, 1463670, 398959, 33630, 2721, 254, 31, 6, 1, 43966297, 10391023, 741721, 49171, 3581, 315, 37, 7, 1, 1496317768, 312129649, 19318376, 1084483

Offset: 0

Henry Bottomley, Dec 31 2002

			Rows start: 1,0,1,10,141,2548,56197,...; 1,1,7,71,1001,18089,398959,...; 1,2,13,132,1861,33630,741721,...; 1,3,19,193,2721,49171,1084483,...; etc.

Rows include A002119 (unsigned), A079165, A001517. Columns include A000012, A001477, A016921 (offset).

T(n, k) =2*T(n-1, k)-T(n-2, k). T(n, k)/T(1, k) tends to ( (n-1)*e - (n-3) )/2 as k increases: e.g. T(3, k)/T(1, k) tends to e.

A072331 a(n) = 2^(n-1) * Sum_{k=0..n} ((n+k)!/(n-k)!)/k!.

Original entry on oeis.org

3, 38, 772, 21768, 786736, 34703456, 1807726656, 108602413184, 7392195003136, 562241229891072, 47257832090862592, 4349969517278922752, 435185983056255725568, 47017486048144734052352

Offset: 1

Benoit Cloitre, Jul 31 2002

Cf. A001517.

Table[2^(n-1) Sum[((n+k)!/(n-k)!)/k!,{k,0,n}],{n,15}] (* Harvey P. Dale, May 22 2012 *)

Definition clarified by Harvey P. Dale, May 22 2012

A360441 Triangle read by rows: T(n,k) is the number of pairs (c,m), where c is a covering of the 1 X (2n) grid with 1 X 2 rectangles and equal numbers of red and blue 1 X 1 squares and m is a matching between red squares and blue squares, such that exactly k matched pairs are adjacent.

Original entry on oeis.org

1, 1, 2, 7, 8, 4, 71, 78, 36, 8, 1001, 1072, 504, 128, 16, 18089, 19090, 9080, 2480, 400, 32, 398959, 417048, 199980, 56960, 10320, 1152, 64, 10391023, 10789982, 5204556, 1523480, 295120, 38304, 3136, 128, 312129649, 322520672, 156264304, 46629632, 9436000, 1336832, 130816, 8192, 256

Offset: 0

William P. Orrick, Mar 08 2023

If row elements are divided by row sums, one obtains a probability distribution that approaches a Poisson distribution with expected value 1 as n approaches infinity.

			Triangle begins:
         1
         1        2
         7        8       4
        71       78      36       8
      1001     1072     504     128     16
     18089    19090    9080    2480    400    32
    398959   417048  199980   56960  10320  1152   64
  10391023 10789982 5204556 1523480 295120 38304 3136 128

Column 1 is |A002119|.
T(n,k) equals 2^k times the corresponding element of the triangle of A168422.
Sum of row n is A001517(n).
Cf. A253667.

def T(n,k):
    return(2^k*sum((-1)^(j-k)*binomial(2*n-j,n)*binomial(n,j)\
     *binomial(j,k)*factorial(n-j) for j in range(k,n+1)))

T(n,k) equals 2^k times the corresponding element of the triangle of A168422.
T(n,k) = 2^k * Sum_{j=k..n} (-1)^(j-k) * C(2*n-j,n) * C(n,j) * C(j,k) * (n-j)!.
Recurrence: T(n,k) = (1/k!) * Sum_{j=0..k} T(n-j,0) * (-1)^j * C(k,j) * Sum_{t=0..min(j,k-j)} (-1)^(j-t) * C(j,t) * (k-j)! / (k-j-t)!
  = (1/k!) * Sum_{j=0..k} T(n-j,0) * (-1)^j * C(k,j) * R(k,j) where R(k,j) is an element of the triangle of A253667.

cp's OEIS Frontend

A168422 Number triangle with row sums given by quadruple factorial numbers A001813.

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A272648 a(n) = A002119(n) mod 7.

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A307495 Expansion of Sum_{k>=0} k!*((1 - sqrt(1 - 4*x))/2)^k.

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A369723 Expansion of e.g.f. exp( 2 * (1-sqrt(1-4*x)) ).

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A369724 Expansion of e.g.f. exp( (5/2) * (1-sqrt(1-4*x)) ).

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A079166 Square array read by antidiagonals of T(n,k)=(4k-2)*T(n,k-1)+T(n,k-2) with T(n,0)=1 and T(n,1)=n.

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A072331 a(n) = 2^(n-1) * Sum_{k=0..n} ((n+k)!/(n-k)!)/k!.

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A360441 Triangle read by rows: T(n,k) is the number of pairs (c,m), where c is a covering of the 1 X (2n) grid with 1 X 2 rectangles and equal numbers of red and blue 1 X 1 squares and m is a matching between red squares and blue squares, such that exactly k matched pairs are adjacent.

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SageMath

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A307495 Expansion of Sum_{k>=0} k!((1 - sqrt(1 - 4x))/2)^k.