A002119 -id:A002119 - cp's OEIS frontend

A360098 Square array read by antidiagonals upwards: T(n,k) is the number of ways of choosing nonnegative numbers for k n-sided dice, k >= 0, n >= 1, so that summing the faces can give any integer from 0 to n^k - 1.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 7, 1, 105, 1, 1, 1, 1, 1, 1, 71, 1, 945, 1, 1, 1, 1, 1, 10, 1, 1001, 1, 10395, 1, 1, 1, 1, 1, 3, 280, 1, 18089, 1, 135135, 1, 1, 1, 1, 1, 7, 15, 15400, 1, 398959, 1

Offset: 1

William P. Orrick, Jan 25 2023

T(n,k) depends on n only through its prime signature. (See A118914.) For example, for fixed k, T(p*q^2,k) will be the same for any pair of distinct primes p and q. Hence we may define T(n,k) = R(s(n),k), where s(n) is the prime signature of n.

Also the number of Krasner factorizations of (x^(n^k)-1) / (x-1) into k polynomials each having n nonzero terms all with coefficient +1. (Krasner and Ranulac, 1937)

			There are 3 ways to assign numbers to two 4-sided dice:
 {{0, 1, 2, 3}, {0, 4, 8, 12}},
 {{0, 1, 8, 9}, {0, 2, 4,  6}},
 {{0, 1, 4, 5}, {0, 2, 8, 10}}.
The northwest corner of T(n,k) begins:
  1   1    1    1       1          1             1 ...     (s(1)  = {})
  1   1    1    1       1          1             1 ...     (s(2)  = {1})
  1   1    1    1       1          1             1 ...     (s(3)  = {1})
  1   1    3   15     105        945         10395 ...     (s(4)  = {2})
  1   1    1    1       1          1             1 ...     (s(5)  = {1})
  1   1    7   71    1001      18089        398959 ...     (s(6)  = {1,1})
  1   1    1    1       1          1             1 ...     (s(7)  = {1})
  1   1   10  280   15400    1401400     190590400 ...     (s(8)  = {3})
  1   1    3   15     105        945         10395 ...     (s(9)  = {2})
  1   1    7   71    1001      18089        398959 ...     (s(10) = {1,1})
  1   1    1    1       1          1             1 ...     (s(11) = {1})
  1   1   42  3660 614040  169200360   69444920160 ...     (s(12) = {1,2})
  1   1    1    1       1          1             1 ...     (s(13) = {1})
  1   1    7   71    1001      18089        398959 ...     (s(14) = {1,1})
  1   1    7   71    1001      18089        398959 ...     (s(15) = {1,1})
  1   1   35 5775 2627625 2546168625 4509264634875 ...     (s(16) = {4})
  ...

M. Krasner and B. Ranulac, Sur une propriété des polynomes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
Matthew C. Lettington and Karl Michael Schmidt, Divisor Functions and the Number of Sum Systems, arXiv:1910.02455 [math.NT], 2019.

For rows of index n = p^j, p prime, or equivalently, for rows of signature {j} we have T(p^2,k) = R({2},k) = A001147(k), T(p^3,k) = R({3},k) = A025035(k), T(p^4,k) = R({4},k) = A025036(k), and, generally, T(p^j,k) = R({j},k) = the k-th element of the j-th column of the square array A060540.
For n = p * q, p and q distinct primes, we have T(p*q,k) = R({1,1},k) = |A002119(k)|.
Column correspondences are T(n,2) = A273013(n) and T(n,3) = A131514(n).
Cf. A118914.

@cached_function
def r(i,M):
    kminus1 = len(M)
    u = tuple([1 for j in range(kminus1)])
    if i > 1 and M == u:
        return(1)
    elif M != u:
        divList = divisors(i)[:-1]
        return(sum(r(M[j],tuple(sorted(M[:j]+tuple([d])+M[j+1:]))) for d in divList for j in range(kminus1)))
def T(n,k):
    if n == 1 or k == 0:
        return(1)
    else:
        return(r(n,tuple([n for j in range(k-1)]))) / factorial(k-1)

Use M to denote a (k-1)-element multiset of positive integers. Let U denote the (k-1)-element multiset whose elements all equal 1 and let N denote the (k-1)-element multiset whose elements all equal n. For i in M, let M_{i,j} denote the result of replacing i with j in M. Then T(1,k) = T(n,0) = 1, while for n > 1 and k > 0 we have T(n,k) = r(n,N) / (k-1)! where r(i,M) is given by the recurrence
   r(i,U) = 1 for i > 1,
   r(i,M) = Sum_{m in M} Sum_{d|i,d

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

Original entry on oeis.org

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

A232691 E.g.f. satisfies: A(x) = exp( 1/A(x)^2 * Integral A(x)^6 dx ).

Original entry on oeis.org

1, 1, 3, 19, 161, 1857, 25843, 433891, 8378913, 185022049, 4565674115, 125075024211, 3755498096257, 122872235056993, 4345683577199283, 165338206044981091, 6730088764152273857, 291935651271257092161, 13440846879808207921027, 654704450541594973156627

Offset: 0

Paul D. Hanna, Dec 06 2013

nonn

Note that G(x) = exp(1/G(x)^2 * Integral G(x)^5 dx) has negative coefficients.

Compare e.g.f. to: B(x) = exp( 1/B(x)^2 * Integral B(x)^2 dx ) where B(y) = Bessel polynomial y_n(-2) (cf. A002119).

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 161*x^4/4! + 1857*x^5/5! +...
Related expansions:
log(A(x)) = x + 2*x^2/2! + 12*x^3/3! + 88*x^4/4! + 976*x^5/5! + 12576*x^6/6! +...
Integral A(x)^6 dx = x + 6*x^2/2! + 48*x^3/3! + 504*x^4/4! + 6576*x^5/5! +...
1/A(x)^2 = 1 - 2*x - 8*x^3/3! - 16*x^4/4! - 384*x^5/5! - 2624*x^6/6! -...

Cf. A232690, A232692.

seq(n! * coeff(series((LambertW(-1, (8*x-3)*exp(-3))/(8*x-3))^(1/4), x, n+1), x, n), n=0..20) # Vaclav Kotesovec, Jan 05 2014

{a(n)=local(A=1+x);for(i=1,n,A=exp(1/A^2*intformal(A^6+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: (LambertW(-1, (8*x-3)*exp(-3))/(8*x-3))^(1/4). - Vaclav Kotesovec, Jan 05 2014

A272647 a(n) = A001517(n) mod 7.

Original entry on oeis.org

1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1, 1, 3, 5, 4, 5, 3, 1

Offset: 0

N. J. A. Sloane, May 09 2016

Periodic with period length 7.

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 (0,0,0,0,0,0,1).

Cf. A001517, A002119, A272648.

Cf. A010876.

f:=proc(n) option remember; if n = 0 then 1 elif n=1 then 3 else f(n-2)+(4*n-2)*f(n-1); fi; end;
[seq(f(n) mod 7, n=0..120)];

PadRight[{},120,{1,3,5,4,5,3,1}] (* Harvey P. Dale, Jul 17 2020 *)

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

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

a(n) = (3*m^6 - 54*m^5 + 365*m^4 - 1140*m^3 + 1582*m^2 - 636*m + 60)/60, where m = n mod 7. - Luce ETIENNE, Oct 18 2018

A303986 Triangle of derivatives of the Niven polynomials evaluated at 0.

Original entry on oeis.org

1, 1, -2, 1, -6, 12, 1, -12, 60, -120, 1, -20, 180, -840, 1680, 1, -30, 420, -3360, 15120, -30240, 1, -42, 840, -10080, 75600, -332640, 665280, 1, -56, 1512, -25200, 277200, -1995840, 8648640, -17297280, 1, -72, 2520, -55440, 831600, -8648640, 60540480, -259459200, 518918400, 1, -90, 3960, -110880, 2162160, -30270240, 302702400, -2075673600, 8821612800, -17643225600, 1, -110, 5940, -205920, 5045040, -90810720, 1210809600, -11762150400, 79394515200, -335221286400, 670442572800

Offset: 0

Wolfdieter Lang, May 07 2018

The Niven potentials N(n, x) = (1/n!)*x^n*(1 - x)^n = Sum_{k=0..n} (-1)^k * x^(n+k)/((n-k)!*k!), with (n-k)!*k! = A098361(n, k), are used to prove the irrationality of Pi^2 (hence Pi). See the Niven and Havil references.

The row polynomials R(n, x) = Sum_{k=0..n} T(n, k) *x^k are R(n, x) = y_n(-2*x), with the Bessel polynomials of Krall and Frink y_n(x) with coefficients given in A001498. There the references are given. - Wolfdieter Lang, May 12 2018

			The triangle T(n, k) begins:
n\k 0   1    2      3      4        5        6          7         8 ...
0:  1
1:  1  -2
2:  1  -6   12
3:  1 -12   60   -120
4:  1 -20  180   -840   1680
5:  1 -30  420  -3360  15120   -30240
6:  1 -42  840 -10080  75600  -332640    66528
7:  1 -56 1512 -25200 277200 -1995840  8648640  -17297280
8:  1 -72 2520 -55440 831600 -8648640 60540480 -259459200 518918400
...

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.
Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.

Muniru A Asiru, Rows n = 0..50

Row sums are A002119.

Cf. A098361, A001498, A113025, A113216.

Flat(List([0..10],n->List([0..n],k->(-1)^k*Binomial(n,n-k)*Factorial(n+k)/Factorial(n)))); # Muniru A Asiru, May 15 2018

T := (n, k) -> (-1)^k*pochhammer(n+1, k)*binomial(n, k):
seq(print(seq(T(n, k), k=0..n)), n=0..9); # Peter Luschny, May 11 2018

T(n,k)=(-1)^k*binomial(n,n-k)*binomial(n+k,n)*k! \\ M. F. Hasler, May 09 2018

T(n, k) = (d/dx)^(n+k) N(n, x) |_{x=0} =: N^{(n+k)}(n, 0), with N(n, x) = (1/n!)*x^n*(1 - x)^n, for n >= 0, k = 0..n.
N^{(n+k)}(n, 1) = (-1)^(n+k)*T(n, k), which has for even n the unsigned rows, and for odd n the unsigned row entries with negative signs.
T(n, k) = (-1)^k*binomial(n, n-k)*((n+k)!/n!).
T(n, k) = (-1)^k*A113025(n,k) with A113025(n,k) = (n+k)!/(k!*(n-k)!) = abs(A113216(n,k)). - M. F. Hasler, May 09 2018
T(n, k) = (-1)^k*Pochhammer(n+1, k)*binomial(n, k). - Peter Luschny, May 11 2018
Recurrence: from the one of the row polynomials R(n, x) = y_n(-2*x): R(n, x) = -2*(2*n-1)*x*R(n-1, x) + R(n-2, x), with R(-1, x) = 1 = R(0, x) = 1, n >= 1 (see A001498), this becomes, for n >= 0, k = 0..n:
  T(n, k) = 0 for n < k, T(n, -1) = 0, T(0, 0) = 1 = T(1, 0) and otherwise
  T(n, k) = -2*(2*n-1)*T(n-1, k-1) + T(n-2, k). - Wolfdieter Lang, May 12 2018

A380261 Expansion of e.g.f. exp( ((1+3*x)^(2/3) - 1)/2 ).

Original entry on oeis.org

1, 1, 0, 2, -14, 146, -1944, 31620, -608068, 13502076, -340052704, 9579145016, -298455813160, 10191129869272, -378469678855904, 15187759126892976, -654936026064200944, 30203464484648818960, -1483333523694819075328, 77291514214052885054496

Offset: 0

Seiichi Manyama, Jan 18 2025

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000085, A002119, A004211, A380262.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(((1+3*x)^(2/3)-1)/2)))

a(n) = Sum_{k=0..n} 3^(n-k) * Stirling1(n,k) * A004211(k) = Sum_{k=0..n} 2^k * 3^(n-k) * Stirling1(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/2)) * 3^n * n! * Sum_{k>=0} binomial(2*k/3,n)/(2^k * k!).

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.

A380262 Expansion of e.g.f. exp( ((1+5*x)^(2/5) - 1)/2 ).

Original entry on oeis.org

1, 1, -2, 16, -206, 3682, -84236, 2348704, -77241380, 2926735516, -125540336024, 6013069027648, -318093606114536, 18418565715581656, -1158626159228481488, 78679416565851286144, -5736477278907382585328, 446936684375920051751440, -37056888825921886749507872

Offset: 0

Seiichi Manyama, Jan 18 2025

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000085, A002119, A380261.

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

a(n) = Sum_{k=0..n} 5^(n-k) * Stirling1(n,k) * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * Stirling1(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

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

A112227 A scaled Hermite triangle.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, 0, -6, 0, 1, 12, 0, -12, 0, 1, 0, 60, 0, -20, 0, 1, -120, 0, 180, 0, -30, 0, 1, 0, -840, 0, 420, 0, -42, 0, 1, 1680, 0, -3360, 0, 840, 0, -56, 0, 1, 0, 15120, 0, -10080, 0, 1512, 0, -72, 0, 1, -30240, 0, 75600, 0, -25200, 0, 2520, 0, -90, 0, 1, 0, -332640, 0, 277200, 0, -55440, 0, 3960, 0, -110, 0, 1, 665280, 0

Offset: 0

Paul Barry, Aug 28 2005

Inverse of number triangle A067147. Diagonal sums are A002119.

			Triangle begins
1;
0,1;
-2,0,1;
0,-6,0,1;
12,0,-12,0,1;
0,60,0,-20,0,1;

Index entries for sequences related to Hermite polynomials

(* The function RiordanArray is defined in A256893. *)
rows = 12;
R = RiordanArray[E^(-#^2)&, #&, rows, True];
R // Flatten

Number triangle T(n, k)=A060821(n, k)/2^k; T(n, k)=n!/(k!*2^((n-k)/2)((n-k)/2)!)*cos(pi*(n-k)/2)*2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1) T(n, k)=A001498((n+k)/2, (n-k)/2)*cos(pi(n-k)/2)*2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1);

Exponential Riordan array (e^(-x^2),x). - Paul Barry, Sep 12 2006

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A086791 Primes found among the numerators of the continued fraction rational approximations to e.

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A360098 Square array read by antidiagonals upwards: T(n,k) is the number of ways of choosing nonnegative numbers for k n-sided dice, k >= 0, n >= 1, so that summing the faces can give any integer from 0 to n^k - 1.

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

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A232691 E.g.f. satisfies: A(x) = exp( 1/A(x)^2 * Integral A(x)^6 dx ).

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A272647 a(n) = A001517(n) mod 7.

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A303986 Triangle of derivatives of the Niven polynomials evaluated at 0.

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A380261 Expansion of e.g.f. exp( ((1+3*x)^(2/3) - 1)/2 ).

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