A111883 -id:A111883 - cp's OEIS frontend

A111595 Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).

1, 0, 1, 1, -2, 1, 0, 9, -6, 1, 9, -36, 42, -12, 1, 0, 225, -300, 130, -20, 1, 225, -1350, 2475, -1380, 315, -30, 1, 0, 11025, -22050, 15435, -4620, 651, -42, 1, 11025, -88200, 220500, -182280, 67830, -12600, 1204, -56, 1, 0, 893025, -2381400, 2302020, -1020600, 235494, -29736, 2052, -72

Offset: 0

Wolfdieter Lang, Aug 23 2005

This is a Sheffer triangle (lower triangular exponential convolution matrix). For Sheffer row polynomials see the S. Roman reference and explanations under A048854.
In the umbral notation of the S. Roman reference this would be called Sheffer for ((sqrt(1-2*t))/(1-t), t/(1-t)).
The associated Sheffer triangle is A111596.
Matrix logarithm equals A112239. - Paul D. Hanna, Aug 29 2005
The row polynomials (1/2^n)* H(n,sqrt(x/2))^2, with the Hermite polynomials H(n,x), have e.g.f. (1/sqrt(1-y^2))*exp(x*y/(1+y)).
The row polynomials s(n,x):=sum(a(n,m)*x^m,m=0..n), together with the associated row polynomials p(n,x) of A111596, satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
The unsigned column sequences are: A111601, A111602, A111777-A111784, for m=1..10.

			The triangle a(n, m) begins:
n\m       0         1         2          3         4         5       6       7     8    9  10 ...
0:        1
1:        0         1
2:        1        -2         1
3:        0         9        -6          1
4:        9       -36        42        -12         1
5:        0       225      -300        130       -20         1
6:      225     -1350      2475      -1380       315       -30       1
7:        0     11025    -22050      15435     -4620       651     -42       1
8:    11025    -88200    220500    -182280     67830    -12600    1204     -56     1
9:        0    893025  -2381400    2302020  -1020600    235494  -29736    2052   -72    1
10:  893025  -8930250  28279125  -30958200  15961050  -4396140  689850  -63000  3285  -90   1
-------------------------------------------------------------------------------------------------

R. P. Boas and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer, 1958, p. 41
S. Roman, The Umbral Calculus, Academic Press, New York, 1984, p. 128.

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

Row sums: A111882. Unsigned row sums: A111883.

Cf. A112239 (matrix log).

row[n_] := CoefficientList[ 1/2^n*HermiteH[n, Sqrt[x/2]]^2, x]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 17 2013 *)

from sympy import hermite, Poly, sqrt, symbols
x = symbols('x')
def a(n): return Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()[::-1]
for n in range(11): print(a(n)) # Indranil Ghosh, May 26 2017

E.g.f. for column m>=0: (1/sqrt(1-x^2))*((x/(1+x))^m)/m!.

a(n, m)=((-1)^(n-m))*(n!/m!)*sum(binomial(2*k, k)*binomial(n-2*k-1, m-1)/(4^k), k=0..floor((n-m)/2)), n>=m>=1. a(2*k, 0)= ((2*k)!/(k!*2^k))^2 = A001818(k), a(2*k+1) = 0, k>=0. a(n, m)=0 if n

A270229 Number of matchings in the 2 X n rook graph P_2 X K_n.

Original entry on oeis.org

1, 2, 7, 32, 193, 1382, 11719, 112604, 1221889, 14639786, 192949639, 2760749048, 42732172993, 709490574158, 12596398359367, 237750425419508, 4757710386662401, 100516614496518866, 2236829315345704711, 52262526676903613264, 1279512810244450887361

Offset: 0

Andrew Howroyd, Mar 13 2016

nonn

Sequence extended to n=0 using closed form. (binomial transform of A111883)

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A270227, A270228, A000085, A081919 (perfect matchings).

a[n_] := Sum[Binomial[n, k]*Abs[HermiteH[k, I/Sqrt[2]]]^2/2^k, {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 01 2017 *)
CoefficientList[Series[E^((2-x)*x/(1-x)) / Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2017 *)

Binomial transform of A111883.
From Vaclav Kotesovec, Oct 01 2017: (Start)
a(n) = (n+1)*a(n-1) + (n-1)^2*a(n-2) - (n-2)*(n-1)^2*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4).
E.g.f.: exp((2-x)*x/(1-x)) / sqrt(1-x^2).
a(n) ~ exp(1/2 + 2*sqrt(n) - n) * n^n / 2.
(End)

A111882 Row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).

Original entry on oeis.org

1, 1, 0, 4, 4, 36, 256, 400, 17424, 784, 1478656, 876096, 154753600, 560363584, 19057250304, 220388935936, 2564046397696, 83038749753600, 327933273309184, 33173161139160064, 26222822450021376, 14475245839622726656

Offset: 0

Wolfdieter Lang, Aug 23 2005

G. C. Greubel, Table of n, a(n) for n = 0..449

Cf. A111883 (unsigned row sums of A111595).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1+x))/Sqrt(1-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 10 2018

With[{nmax = 50}, CoefficientList[Series[Exp[x/(1 + x)]/Sqrt[1 - x^2], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)

x='x+O('x^30); Vec(serlaplace(exp(x/(1+x))/sqrt(1-x^2))) \\ G. C. Greubel, Jun 10 2018

from sympy import hermite, Poly, sqrt
def a(n): return sum(Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()) # Indranil Ghosh, May 26 2017

E.g.f.: exp(x/(1+x))/sqrt(1-x^2).
a(n) = Sum_{m=0..n} A111595(n, m), n>=0.
A111882(n) = A001464(n)^2. - Mark van Hoeij, Nov 11 2009
D-finite with recurrence a(n) +(n-2)*a(n-1) -(n-1)*(n-2)*a(n-2) -(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Oct 05 2014

A173869 Irregular table T(n,k) = A164341(n,k) * A036039(n,k) read by rows.

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 24, 24, 18, 24, 10, 120, 120, 120, 120, 90, 80, 26, 720, 720, 720, 480, 720, 720, 300, 480, 540, 300, 76, 5040, 5040, 5040, 5040, 5040, 5040, 3360, 3780, 3360, 5040, 2100, 2100, 2520, 1092, 232, 40320, 40320, 40320, 40320, 25200, 40320

Offset: 1

Alford Arnold, Mar 12 2010

The n-th row has A000041(n) columns.
The row sums yield A111883(n) = A000085(n)^2.
A000041 and A000085 are also relevant to the table defined by A104778.

			The row lengths of A164341 and A036039 are the same, so one can multiply
the flattened arrays point-by-point to compute this sequence here:
1..2..2..3..2..4.. A164341 times
1..1..1..2..3..1.. A036039 yields
1..2..2..6..6..4..

Cf. A000041, A000085, A036039, A104778, A111883, A164341.

Definition rephrased by R. J. Mathar, Mar 26 2010

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A111595 Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).

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A270229 Number of matchings in the 2 X n rook graph P_2 X K_n.

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A111882 Row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).

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A173869 Irregular table T(n,k) = A164341(n,k) * A036039(n,k) read by rows.

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