A038207 -id:A038207 - cp's OEIS frontend

A053117 Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).

1, 0, 2, -1, 0, 4, 0, -4, 0, 8, 1, 0, -12, 0, 16, 0, 6, 0, -32, 0, 32, -1, 0, 24, 0, -80, 0, 64, 0, -8, 0, 80, 0, -192, 0, 128, 1, 0, -40, 0, 240, 0, -448, 0, 256, 0, 10, 0, -160, 0, 672, 0, -1024, 0, 512, -1, 0, 60, 0, -560, 0, 1792, 0, -2304, 0, 1024, 0, -12, 0, 280, 0, -1792, 0, 4608, 0, -5120, 0, 2048, 1, 0, -84, 0, 1120, 0, -5376, 0, 11520, 0, -11264, 0, 4096

Offset: 0

Wolfdieter Lang

G.f. for row polynomials U(n,x) (signed triangle): 1/(1-2*x*z+z^2). Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,2*x) as row polynomials with g.f. 1/(1-2*x*z-z^2).
Row sums (unsigned triangle) A000129(n+1) (Pell). Row sums (signed triangle) A000027(n+1) (natural numbers).
The o.g.f. for the Legendre polynomials L(n,x) is 1 / sqrt(1- 2x*z + z^2), and squaring it gives the o.g.f. of this entry, so Sum_{k=0..n} L(k,x) L(n-k,x) = U(n,x). This reduces to U(n,x) = L(n/2,x)^2 + 2*Sum_{k=0...n/2-1} L(k,x) L(n-k,x) for n even and U(n,x) = 2*Sum_{k=0..(n-1)/2} L(k,x) L(n-k.x) for odd n. (Cf. also Allouche et al.) For a connection through the Legendre polynomials to elliptic curves and modular forms, see the MathOverflow question below. For the normalized Legendre polynomials, see A100258. (Cf. A097610 with h1 = -2x and h2 = 1, A207538, A099089 and A133156.) - Tom Copeland, Feb 04 2016
The compositional inverse of the shifted o.g.f. x / (1 + 2xz + z^2) for differently signed row polynomials of this entry is the shifted o.g.f. of A121448. The unsigned, non-vanishing antidiagonals (top to bottom) of this triangle are the rows of A038207. - Tom Copeland, Feb 08 2016

			Triangle begins:
   1;
   0,  2;
  -1,  0,   4;
   0, -4,   0, 8;
   1,  0, -12, 0, 16;
  ...
E.g., fourth row (n=3) {0,-4,0,8} corresponds to polynomial U(3,x) = -4*x + 8*x^3.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 22, page 196.

T. D. Noe, Rows n=0..100 of triangle, flattened
J.-P. Allouche and G. Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol. 214, Issue 1-3, 21 March 2000, pp. 21-49.
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 5.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). - From _Tom Copeland_, Oct 11 2014
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
MathOverflow, Geometric picture of invariant differential of an elliptic curve, Dec 4 2011.
Valentin Ovsienko, Towards quantized complex numbers: q-deformed Gaussian integers and the Picard group, arXiv:2103.10800 [math.QA], 2021.
R. Pemantle and M. C. Wilson, Asymptotics of multivariate sequences, I: smooth points of the singular variety, arXiv:math/0003192 [math.CO], 2000.
A. Sapounakis, I. Tasoulas, and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
Index entries for sequences related to Chebyshev polynomials.

Cf. A000027, A000129, A049310, A053118.

Cf. A038207, A097610, A099089, A100258, A121448, A133156, A207538.

using Nemo
function A053117Row(n)
    R, x = PolynomialRing(ZZ, "x")
    p = chebyshev_u(n, x)
    [coeff(p, j) for j in 0:n] end
for n in 0:6 A053117Row(n) |> println end # Peter Luschny, Mar 13 2018

seq(seq(coeff(orthopoly[U](n,x),x,j),j=0..n),n=0..16); # Robert Israel, Feb 09 2016

Flatten[ Table[ CoefficientList[ ChebyshevU[n, x], x], {n, 0, 12}]](* Jean-François Alcover, Nov 24 2011 *)

T(n, k) = polcoeff(polchebyshev(n,2), k); \\ Michel Marcus, Feb 10 2016

a(n, m) = (2^m)*A049310(n,m).

a(n, m) := 0 if n

If n and k are of the same parity then a(n,k)=(-1)^((n-k)/2)*sum(binomial((n+k)/2,i)*binomial((n+k)/2-i,(n-k)/2),i=0..k) and a(n,k)=0 otherwise. - Milan Janjic, Apr 13 2008

A130321 Triangle, (2^0, 2^1, 2^2, ...) in every column.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 4, 2, 1, 16, 8, 4, 2, 1, 32, 16, 8, 4, 2, 1, 64, 32, 16, 8, 4, 2, 1, 128, 64, 32, 16, 8, 4, 2, 1, 256, 128, 64, 32, 16, 8, 4, 2, 1, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1

Offset: 0

Gary W. Adamson, May 24 2007

A130321^2 = A130322. Binomial transform of A130321 = triangle A027649. A007318^2 = A038207 = A007318(n,k) * A130321(n,k); i.e., the square of Pascal's triangle = dot product of Pascal's triangle rows and A130321 rows: A007318^2 = (1; 2,1; 4,4,1; 8,12,6,1;...), where row 3, (8,12,6,1) = (1,3,3,1) dot (8,4,2,1).
Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A130321 is the reverse reluctant sequence of sequence of power of 2 (A000079). - Boris Putievskiy, Dec 13 2012
From Wolfdieter Lang, Jan 10 2015: (Start)
This is the Riordan array (1/(1-2*x), x).
Row sums give A000225(n+1) = 2^(n+1) - 1.
Alternating row sums give A001045(n+1).
The inverse Riordan array is (1-2*x, x) = A251635. (End)

			The triangle T(n,m) begins:
  n\m     0   1   2   3  4  5  6  7  8  9 10 ...
  0:      1
  1:      2   1
  2:      4   2   1
  3:      8   4   2   1
  4:     16   8   4   2  1
  5:     32  16   8   4  2  1
  6:     64  32  16   8  4  2  1
  7:    128  64  32  16  8  4  2  1
  8:    256 128  64  32 16  8  4  2  1
  9:    512 256 128  64 32 16  8  4  2  1
 10:   1024 512 256 128 64 32 16  8  4  2  1
 ... Reformatted. - _Wolfdieter Lang_, Jan 10 2015

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A059268, A027649, A130322, A038207, A131816, A000225, A001045, A251635.

a130321 n k = a130321_tabl !! n !! k
a130321_row n = a130321_tabl !! n
a130321_tabl = iterate (\row -> (2 * head row) : row) [1]
-- Reinhard Zumkeller, Feb 27 2013

T[n_, m_] := 2^(n-m);
Table[T[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

Triangle, (1, 2, 4, 8, ...) in every column. Rows are reversals of A059268 terms.
a(n)=2^m, where m=(t*t + 3*t + 4)/2 - n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012
From Wolfdieter Lang, Jan 10 2015: (Start)
T(n, m) = 2^(n-m) if n >= m >= 0 and 0 otherwise.
G.f. of row polynomials R(n,x) = Sum_{m=0..n} 2^(n-m)*x^m is 1/((1-2*z)*(1-x*z)) (Riordan property).
G.f. column m (with leading zeros) x^m/(1-2*x), m >= 0.
The diagonal sequences are D(k) = repeat(2^k), k >= 0. (End)

More terms from Philippe Deléham, Feb 08 2009

A238363 Coefficients for the commutator for the logarithm of the derivative operator [log(D),x^n D^n]=d[(xD)!/(xD-n)!]/d(xD) expanded in the operators :xD:^k.

Original entry on oeis.org

1, -1, 2, 2, -3, 3, -6, 8, -6, 4, 24, -30, 20, -10, 5, -120, 144, -90, 40, -15, 6, 720, -840, 504, -210, 70, -21, 7, -5040, 5760, -3360, 1344, -420, 112, -28, 8, 40320, -45360, 25920, -10080, 3024, -756, 168, -36, 9, -362880, 403200, -226800, 86400, -25200, 6048, -1260, 240, -45, 10

Offset: 1

Tom Copeland, Feb 25 2014

Let D=d/dx and [A,B]=A·B-B·A. Then each row corresponds to the coefficients of the operators :xD:^k = x^k D^k in the expansion of the commutator [log(D),:xD:^n]=[-log(x),:xD:^n]=sum(k=0 to n-1, a(n,k) :xD:^k). The e.g.f. is derived from [log(D), exp(t:xD:)]=[-log(x), exp(t:xD:)]= log(1+t)exp(t:xD:), using the shift property exp(t:xD:)f(x)=f((1+t)x).
The reversed unsigned array is A111492.
See the mathoverflow link and link therein to an associated mathstackexchange question for other formulas for log(D). In addition, R_x = log(D) = -log(x) + c - sum[n=1 to infnty, (-1)^n 1/n :xD:^n/n!]=
-log(x) + Psi(1+xD) = -log(x) + c + Ein(:xD:), where c is the Euler-Mascheroni constant, Psi(x), the digamma function, and Ein(x), a breed of the exponential integrals (cf. Wikipedia). The :xD:^k ops. commute; therefore, the commutator reduces to the -log(x) term.
Also the n-th row corresponds to the expansion of d[(xD)!/(xD-n)!]/d(xD) = d[:xD:^n]/d(xD) in the operators :xD:^k, or, equivalently, the coefficients of x in  d[z!/(z-n)!]/dz=d[St1(n,z)]]/dz evaluated umbrally with z=St2(.,x), i.e., z^n replaced by St2(n,x), where St1(n,x) and St2(n,x) are the signed and unsigned Stirling polynomials of the first (A008275) and second (A008277) kinds. The derivatives of the unsigned St1 are A028421. See examples. This formalism follows from the relations between the raising and lowering operators presented in the MathOverflow link and the Pincherle derivative. The results can be generalized through the operator relations in A094638, which are related to the celebrated Witt Lie algebra and pseudodifferential operators / symbols, to encompass other integral arrays.
A002741(n)*(-1)^(n+1) (row sums), A002104(n)*(-1)^(n+1) (alternating row sums). Column sequences: A133942(n-1), A001048(n-1), A238474, ... - Wolfdieter Lang, Mar 01 2014
Add an additional head row of zeros to the lower triangular array and denote it as T (with initial indexing in columns and rows being 0). Let dP = A132440, the infinitesimal generator for the Pascal matrix, and I, the identity matrix, then exp(T)=I+dP, i.e., T=log(I+dP). Also, (T_n)^n=0, where T_n denotes the n X n submatrix, i.e., T_n is nilpotent of order n. - Tom Copeland, Mar 01 2014
Any pair of lowering and raising ops. L p(n,x) = n·p(n-1,x) and R p(n,x) = p(n+1,x) satisfy [L,R]=1 which implies (RL)^n = St2(n,:RL:), and since (St2(·,u))!/(St2(·,u)-n)!= u^n, when evaluated umbrally, d[(RL)!/(RL-n)!]/d(RL) = d[:RL:^n]/d(RL) is well-defined and gives A238363 when the LHS is reduced to a sum of :RL:^k terms, exactly as for L=d/dx and R=x above. (Note that R_x above is a raising op. different from x, with associated L_x=-xD.) - Tom Copeland, Mar 02 2014
For relations to colored forests, disposition of flags on flagpoles, and the colorings of the vertices of the complete graphs K_n, encoded in their chromatic polynomials, see A130534. - Tom Copeland, Apr 05 2014
The unsigned triangle, omitting the main diagonal, gives A211603. See also A092271. Related to the infinitesimal generator of A008290. - Peter Bala, Feb 13 2017

			The first few row polynomials are
p(1,x)=  1
p(2,x)= -1 + 2x
p(3,x)=  2 - 3x + 3x^2
p(4,x)= -6 + 8x - 6x^2 + 4x^3
p(5,x)= 24 -30x +20x^2 -10x^3 + 5x^4
...........
For n=3: z!/(z-3)!=z^3-3z^2+2z=St1(3,z) with derivative 3z^2-6z+2, and
3·St2(2,x)-6·St2(1,x)+2=3(x^2+x)-6x+2=3x^2-3x+2=p(3,x). To see the relation to the operator formalism, note that (xD)^k=St2(k,:xD:) and (xD)!/(xD-k)!=[St2(·,:xD:)]!/[St2(·,:xD:)-k]!= :xD:^k.
The triangle a(n,k) begins:
n\k       0       1       2      3      4     5      6    7   8   9 ...
1:        1
2:       -1       2
3:        2      -3       3
4:       -6       8      -6      4
5:       24     -30      20    -10      5
6:     -120     144     -90     40    -15     6
7:      720    -840     504   -210     70   -21      7
8:    -5040    5760   -3360   1344   -420   112    -28    8
9:    40320  -45360   25920 -10080   3024  -756    168  -36   9
10: -362880  403200 -226800  86400 -25200  6048  -1260  240 -45  10
... formatted by _Wolfdieter Lang_, Mar 01 2014
-----------------------------------------------------------------------

Tom Copeland, Riemann zeta function at positive integers and an Appell sequence of polynomials, 2012.
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator, 2014.
Tom Copeland, Bernoulli, Blissard, and Lie meet Stirling and the simplices: State number operators and normal ordering, 2014.
Tom Copeland, Compositional inverse operators and Sheffer sequences, 2016.

Cf. A094587, A132013, A132014, A132159, A235706, A238385.

Cf. A008290, A092271, A211603.

a[n_, k_] := (-1)^(n-k-1)*n!/((n-k)*k!); Table[a[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 09 2015 *)

a(n,k) = (-1)^(n-k-1)*n!/((n-k)*k!) for k=0 to (n-1).
E.g.f.: log(1+t)*exp(x*t).
E.g.f.for unsigned array: -log(1-t)*exp(x*t).
The lowering op. for the row polynomials is L=d/dx, i.e., L p(n,x) = n*p(n-1,x).
An e.g.f. for an unsigned related version is -log(1+t)*exp(x*t)/t= exp(t*s(·,x)) with s(n,x)=(-1)^n * p(n+1,-x)/(n+1). Let L=d/dx and R= x-(1/((1-D)log(1-D))+1/D),then R s(n,x)= s(n+1,x) and L s(n,x)= n*s(n-1,x), defining a special Sheffer sequence of polynomials, an Appell sequence. So, R (-1)^(n-1) p(n,-x)/n = (-1)^n p(n+1,-x)/(n+1).
From Tom Copeland, Apr 17 2014: (Start)
Dividing each diagonal by its first element (-1)^(n-1)*(n-1)! yields the reverse of A104712.
Multiply each n-th diagonal of the Pascal lower triangular matrix by x^n and designate the result as A007318(x) = P(x). Then with dP = A132440, M = padded A238363 = A238385-I, I = identity matrix, and (B(.,x))^n = B(n,x) = the n-th Bell polynomial Bell(n,x) of A008277,
A) P(x)= exp(x*dP) = exp[x*(e^M-I)] = exp[M*B(.,x)] = (I+dP)^B(.,x), and
B) P(:xD:)=exp(dP:xD:)=exp[(e^M-I):xD:]=exp[M*B(.,:xD:)]=exp[M*xD]=
  (1+dP)^(xD) with action P(:xD:)g(x) = exp(dP:xD:)g(x) = g[(I+dP)*x].
C) P(x)^m = P(m*x). P(2x) = A038207(x) = exp[M*B(.,2x)], face vectors of n-D hypercubes. (End)
From Tom Copeland, Apr 26 2014: (Start)
M = padded A238363 = A238385-I
A)  = [St1]*[dP]*[St2] = [padded A008275]*A132440*A048993
B)  = [St1]*[dP]*[St1]^(-1)
C)  = [St2]^(-1)*[dP]*[St2]
D)  = [St2]^(-1)*[dP]*[St1]^(-1),
  where [St1]=padded A008275 just as [St2]=A048993=padded A008277.
E) P(x) = [St2]*exp(x*M)*[St1] = [St2]*(I + dP)^x*[St1].
F) exp(x*M) = [St1]*P(x)*[St2] = (I + dP)^x,
  where (I + dP)^x = sum(k>=0, C(x,k)*dP^k).
Let the row vector Rv=(c0 c1 c2 c3 ...) and the column vector Cv(x)=(1 x x^2 x^3 ...)^Transpose. Form the power series V(x)= Rv * Cv(x) and W(y) := V(x.) evaluated umbrally with (x.)^n = x_n = (y)_n = y!/(y-n)!. Then
G) U(:xD:) = dV(:xD:)/d(xD) = dW(xD)/d(xD) evaluated with (xD)^n = Bell(n,:xD:),
H) U(x) = dV(x.)/dy := dW(y)/dy evaluated with y^n=y_n=Bell(n,x), and
I) U(x) = Rv * M * Cv(x).   (Cf. A132440, A074909.) (End)
The Bernoulli polynomials Ber_n(x) are related to the polynomials q_n(x) = p(n+1,x) / (n+1) with the e.g.f. [log(1+t)/t] e^(xt) (cf. s_n (x) above) as Ber_n(x) = St2_n[q.(St1.(x))], umbrally, or [St2]*[q]*[St1], in matrix form. Since q_n(x) is an Appell sequence of polynomials, q_n(x) = [log(1+D_x)/D_x]x^n. - Tom Copeland, Nov 06 2016

Pincherle formalism added by Tom Copeland, Feb 27 2014

A002409 a(n) = 2^n*C(n+6,6). Number of 6D hypercubes in an (n+6)-dimensional hypercube.

Original entry on oeis.org

1, 14, 112, 672, 3360, 14784, 59136, 219648, 768768, 2562560, 8200192, 25346048, 76038144, 222265344, 635043840, 1778122752, 4889837568, 13231325184, 35283533824, 92851404800, 241413652480, 620777963520, 1580162088960

Offset: 0

N. J. A. Sloane

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>5, a(n-6) is equal to the number of (n+6)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Herbert Izbicki, Über Unterbaeume eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
Herbert Izbicki, Über Unterbaeume eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

First differences are in A006976.
Cf. A000079, A001787, A001788, A001789, A003472, A054849, A054851, A082140.
a(n) = A038207(n+6,6).

[2^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011

A002409:=-1/(2*z-1)**7; # Simon Plouffe in his 1992 dissertation
seq(binomial(n+6,6)*2^n,n=0..22); # Zerinvary Lajos, Jun 16 2008

CoefficientList[Series[1/(1-2x)^7,{x,0,40}],x] (* or *) LinearRecurrence[ {14,-84,280,-560,672,-448,128},{1,14,112,672,3360,14784,59136},40] (* Harvey P. Dale, Jan 24 2022 *)

G.f.: 1/(1-2*x)^7.
a(n) = 2*a(n-1) + A054849(n-1).
For n>0, a(n) = 2*A082140(n).
a(n) = Sum_{i=6..n+6} binomial(i,6)*binomial(n+6,i). Example: for n=5, a(5) = 1*462 + 7*330 + 28*165 + 84*55 + 210*11 + 462*1 = 14784. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 47/5 - 12*log(2).
Sum_{n>=0} (-1)^n/a(n) = 2916*log(3/2) - 5907/5. (End)
n*a(n) +2*(-n-6)*a(n-1)=0. - R. J. Mathar, Jul 22 2025

More terms from Henry Bottomley and James Sellers, Apr 15 2000

Typo in definition corrected by Zerinvary Lajos, Jun 16 2008

A038231 Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).

Original entry on oeis.org

1, 4, 1, 16, 8, 1, 64, 48, 12, 1, 256, 256, 96, 16, 1, 1024, 1280, 640, 160, 20, 1, 4096, 6144, 3840, 1280, 240, 24, 1, 16384, 28672, 21504, 8960, 2240, 336, 28, 1, 65536, 131072, 114688, 57344, 17920, 3584, 448, 32, 1, 262144, 589824, 589824, 344064, 129024, 32256, 5376, 576, 36, 1

Offset: 0

N. J. A. Sloane

Triangle of coefficients in expansion of (4+x)^n. - N-E. Fahssi, Apr 13 2008

			Triangle begins:
      1;
      4,      1;
     16,      8,      1;
     64,     48,     12,     1;
    256,    256,     96,    16,     1;
   1024,   1280,    640,   160,    20,    1;
   4096,   6144,   3840,  1280,   240,   24,   1;
  16384,  28672,  21504,  8960,  2240,  336,  28,  1;
  65536, 131072, 114688, 57344, 17920, 3584, 448, 32, 1;

Indranil Ghosh, Rows 0..125 of triangle, flattened
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A000302, A013611 (row-reversed), A000351 (row sums).

Flat(List([0..10], n-> List([0..n], k-> 4^(n-k)*Binomial(n, k) ))); # G. C. Greubel, Jul 20 2019

[4^(n-k)*Binomial(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 20 2019

for i from 0 to 10 do seq(binomial(i, j)*4^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 4^(n-1)); # Peter Luschny, Oct 09 2022

Table[4^(n-k)*Binomial[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 20 2019 *)

T(n,k) = 4^(n-k)*binomial(n, k); \\ G. C. Greubel, Jul 20 2019

[[4^(n-k)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 20 2019

G.f. for j-th column is (x^j)/(1-4*x)^(j+1).
Convolution triangle of A000302 (powers of 4).
Sum_{k=0..n} T(n,k)*(-1)^k*A000108(k) = A001700(n). - Philippe Deléham, Nov 27 2009
See A038207 and A027465 and replace 2 and 3 in analogous formulas with 4. - Tom Copeland, Oct 26 2012

A039683 Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).

Original entry on oeis.org

1, -2, 1, 8, -6, 1, -48, 44, -12, 1, 384, -400, 140, -20, 1, -3840, 4384, -1800, 340, -30, 1, 46080, -56448, 25984, -5880, 700, -42, 1, -645120, 836352, -420224, 108304, -15680, 1288, -56, 1, 10321920, -14026752, 7559936, -2153088, 359184, -36288, 2184, -72, 1

Offset: 1

Wouter Meeussen

T(n,m) = R_n^m(a=0,b=2) in the notation of the given reference.
Exponential Riordan array [1/(1+2x),log(1+2x)/2]. The unsigned triangle is [1/(1-2x),log(1/sqrt(1-2x))]. - Paul Barry_, Apr 29 2009
The n-th row is related to the expansion of z^(-2n)*(z^3 d/dz)^n in polynomials of the Euler operator D=(z d/dz). E.g., z^(-6)(z^3 d/dz)^3 = D^3 + 6 D^2 + 8 D. See Copeland link for relations to Bell / Exponential / Touchard polynomial operators. - Tom Copeland, Nov 14 2013
A refinement of this array is given by A231846. - Tom Copeland, Nov 15 2013
Also the Bell transform of the double factorial of even numbers A000165 except that the values are unsigned and in addition a first column (1,0,0 ...) is added on the left side of the triangle. For the Bell transform of the double factorial of odd numbers A001147 see A132062. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015
The signed triangle is also the inverse Bell transform of A000079 (see Luschny link). - John Keith, Nov 24 2020

			Triangle starts:
  {1},
  {2,1},
  {8,6,1},
  {48,44,12,1},
  ...
From _Paul Barry_, Apr 29 2009: (Start)
The unsigned triangle [1/(1-2x),log(1/sqrt(1-2x))] has production matrix:
  2, 1,
  4, 4, 1,
  8, 12, 6, 1,
  16, 32, 24, 8, 1,
  32, 80, 80, 40, 10, 1,
  64, 192, 240, 160, 60, 12, 1
which is A007318^{2} beheaded. (End)

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Tom Copeland, Addendum to Mathemagical Forests.
P. Feijão, F. V. Martinez, and A. Thévenin, On the distribution of cycles and paths in multichromosomal breakpoint graphs and the expected value of rearrangement distance, BMC Bioinformatics 16:Suppl19 (2015), S1. doi:10.1186/1471-2105-16-S19-S1
Lisa Glaser, Causal set actions in various dimensions, J. Phys.: Conf. Ser. 306 (2011), 012041.
Wolfdieter Lang, First 9 rows and comment.
Peter Luschny, The Bell transform
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres relies aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

First column (unsigned triangle) is (2(n-1))!! = 1, 2, 8, 48, 384...= A000165(n-1) and the row sums (unsigned) are (2n-1)!! = 1, 3, 15, 105, 945... = A001147(n-1).
Cf. A051141, A051142.
Cf. A000165, A132062, A264428.
Cf. A038207.

Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 6} ]

# uses[bell_transform from A264428]
# Unsigned values and an additional first column (1,0,0,...).
def A039683_unsigned_row(n):
    a = sloane.A000165
    dblfact = a.list(n)
    return bell_transform(n, dblfact)
[A039683_unsigned_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015

T(n, m) = T(n-1, m-1) - 2*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) := 0, n

E.g.f. for m-th column of signed triangle: (((log(1+2*x))/2)^m)/m!.

E.g.f.: (1+2*x)^(y/2). O.g.f. for n-th row of signed triangle: Sum_{m=0..n} Stirling1(n, m)*2^(n-m)*x^m. - Vladeta Jovovic, Feb 11 2003

T(n, m) = S1(n, m)*2^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

The production matrix below is A038207 with the first row removed. With the initial index n = 0, the associated differential raising operator is R = e^(2D)*x = (2+x)*e^(2D) with D = d/dx, i.e., R p_n(x) = p_(n+1)(x) where p_n(x) is the n-th unsigned row polynomial and p_0(x) = 1, so p_(n+1)(x) = (2+x) * p_n(2+x). - Tom Copeland, Oct 11 2016

Additional comments from Wolfdieter Lang

Title revised by Tom Copeland, Dec 21 2013

A054849 a(n) = 2^(n-5)*binomial(n,5). Number of 5D hypercubes in an n-dimensional hypercube.

Original entry on oeis.org

1, 12, 84, 448, 2016, 8064, 29568, 101376, 329472, 1025024, 3075072, 8945664, 25346048, 70189056, 190513152, 508035072, 1333592064, 3451650048, 8820883456, 22284337152, 55710842880, 137950658560, 338606161920

Offset: 5

Henry Bottomley, Apr 14 2000

With 5 leading zeros, binomial transform of binomial(n,5). - Paul Barry, Apr 10 2003

If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>4, a(n) is equal to the number of (n+5)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

G. C. Greubel, Table of n, a(n) for n = 5..1000
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (12, -60, 160, -240, 192, -64).

Cf. A000079, A001787, A001788, A001789, A003472, A002409, A054851.
a(n) = A038207(n,5).
Equals 2 * A082139. First differences are in A006975.

List([5..30], n-> 2^(n-5)*Binomial(n,5)); # G. C. Greubel, Aug 27 2019

[2^(n-5)*Binomial(n,5): n in [5..30]]; // G. C. Greubel, Aug 27 2019

seq(binomial(n+5,5)*2^n,n=0..22); # Zerinvary Lajos, Jun 13 2008

Table[2^(n-5)*Binomial[n,5], {n,5,30}] (* G. C. Greubel, Aug 27 2019 *)

vector(25, n, 2^(n-1)*binomial(n+4,5)) \\ G. C. Greubel, Aug 27 2019

[lucas_number2(n, 2, 0)*binomial(n,5)/32 for n in range(5, 28)] # Zerinvary Lajos, Mar 10 2009

a(n) = 2*a(n-1) + A003472(n-1).
From Paul Barry, Apr 10 2003: (Start)
O.g.f.: x^5/(1-2*x)^6.
E.g.f.: exp(2*x)*(x^5/5!) (with 5 leading zeros). (End)
a(n) = Sum_{i=5..n} binomial(i,5)*binomial(n,i). Example: for n=8, a(8) = 1*56 + 6*28 + 21*8 + 56*1 = 448. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=5} 1/a(n) = 10*log(2) - 35/6.
Sum_{n>=5} (-1)^(n+1)/a(n) = 810*log(3/2) - 655/2. (End)

More terms from James Sellers, Apr 15 2000

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