A163944 -id:A163944 - cp's OEIS frontend

A163940 Triangle related to the divergent series 1^m1! - 2^m2! + 3^m3! - 4^m4! + ... for m >= -1.

1, 1, 0, 1, 2, 0, 1, 5, 3, 0, 1, 9, 17, 4, 0, 1, 14, 52, 49, 5, 0, 1, 20, 121, 246, 129, 6, 0, 1, 27, 240, 834, 1039, 321, 7, 0, 1, 35, 428, 2250, 5037, 4083, 769, 8, 0, 1, 44, 707, 5214, 18201, 27918, 15274, 1793, 9, 0, 1, 54, 1102, 10829, 54111, 133530, 145777, 55152, 4097, 10, 0

Offset: 0

Johannes W. Meijer, Aug 13 2009

The divergent series g(x,m) = Sum_{k >= 1} (-1)^(k+1)*k^m*k!*x^k, m >= -1, are related to the higher order exponential integrals E(x,m,n=1), see A163931.
Hardy, see the link below, describes how Euler came to the rather surprising conclusion that g(x,-1) = exp(1/x)*Ei(1,1/x) with Ei(1,x) = E(x,m=1,n=1). From this result it follows inmediately that g(x,0) = 1 - g(x,-1). Following in Euler's footsteps we discovered that g(x,m) = (-1)^(m) * (M(x,m)*x - ST(x,m)* Ei(1,1/x) * exp(1/x))/x^(m+1), m => -1.
So g(x=1,m) = (-1)^m*(A040027(m) - A000110 (m+1)*A073003), with A040027(m = -1) = 0. We observe that A073003 = - exp(1)*Ei(-1) is Gompertz's constant, A000110 are the Bell numbers and A040027 was published a few years ago by Gould.
The polynomial coefficients of the M(x,m) = Sum_{k = 0..m} a(m,k) * x^k, for m >= 0, lead to the triangle given above. We point out that M(x,m=-1) = 0.
The polynomial coefficients of the ST(x,m) = Sum_{k = 0..m+1} S2(m+1, k) * x^k, m >= -1, lead to the Stirling numbers of the second kind, see A106800.
The formulas that generate the coefficients of the left hand columns lead to the Minkowski numbers A053657. We have a closer look at them in A163972.
The right hand columns have simple generating functions, see the formulas. We used them in the first Maple program to generate the triangle coefficients (n >= 0 and 0 <= k <= n). The second Maple program calculates the values of g(x,m) for m >= -1, at x=1.

			The first few triangle rows are:
  [1]
  [1, 0]
  [1, 2, 0]
  [1, 5, 3, 0]
  [1, 9, 17, 4, 0]
  [1, 14, 52, 49, 5, 0]
The first few M(x,m) are:
  M(x,m=0) = 1
  M(x,m=1) = 1 + 0*x
  M(x,m=2) = 1 + 2*x + 0*x^2
  M(x,m=3) = 1 + 5*x + 3*x^2 + 0*x^3
The first few ST(x,m) are:
  ST(x,m=-1) = 1
  ST(x,m=0) = 1 + 0*x
  ST(x,m=1) = 1 + 1*x + 0*x^2
  ST(x,m=2) = 1 + 3*x + x^2 + 0*x^3
  ST(x,m=3) = 1 + 6*x + 7*x^2 + x^3 + 0*x^4
The first few g(x,m) are:
  g(x,-1) = (-1)*(- (1)*Ei(1,1/x)*exp(1/x))/x^0
  g(x,0) = (1)*((1)*x - (1)*Ei(1,1/x)*exp(1/x))/x^1
  g(x,1) = (-1)*((1)*x - (1+ x)*Ei(1,1/x)*exp(1/x))/x^2
  g(x,2) = (1)*((1+2*x)*x - (1+3*x+x^2)*Ei(1,1/x)*exp(1/x))/x^3
  g(x,3) = (-1)*((1+5*x+3*x^2)*x - (1+6*x+7*x^2+x^3)*Ei(1,1/x)*exp(1/x))/x^4

G. H. Hardy, Divergent Series, Oxford University Press, 1949. pp. 26-29 and pp. 7-8.
Maurice de Gosson, Branko Dragovich and Andrei Khrennikov, Some p-adic differential equations, (see Section 5), arxiv:math-ph/0010023, Oct 2000.

The row sums equal A040027 (Gould).
A000007, A000027, A000337, A163941 and A163942 are the first five right hand columns.
A000012, A000096, A163943 and A163944 are the first four left hand columns.
Cf. A163931, A163972, A106800 (Stirling2), A000110 (Bell), A073003 (Gompertz), A053657 (Minkowski), A014619.

nmax := 10; for p from 1 to nmax do Gf(p) := convert(series(1/((1-(p-1)*x)^2*product((1-k1*x), k1=1..p-2)), x, nmax+1-p), polynom); for q from 0 to nmax-p do a(p+q-1, q) := coeff(Gf(p), x, q) od: od: seq(seq(a(n, k), k=0..n), n=0..nmax-1);
# End program 1
nmax1:=nmax; A040027 := proc(n): if n = -1 then 0 elif n= 0 then 1 else add(binomial(n, k1-1)*A040027(n-k1), k1 = 1..n) fi: end: A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1, i) * A000110(n-1-i), i=0..n-1); fi; end: A073003 := - exp(1) * Ei(-1): for n from -1 to nmax1 do g(1, n) := (-1)^n * (A040027(n) - A000110(n+1) * A073003) od;
# End program 2

nmax = 11;
For[p = 1, p <= nmax, p++, gf = 1/((1-(p-1)*x)^2*Product[(1-k1*x), {k1, 1, p-2}]) + O[x]^(nmax-p+1) // Normal; For[q = 0, q <= nmax-p, q++, a[p+q-1, q] = Coefficient[gf, x, q]]];
Table[a[n, k], {n, 0, nmax-1}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 02 2019, from 1st Maple program *)

The generating functions of the right hand columns are Gf(p, x) = 1/((1 - (p-1)*x)^2 * Product_{k = 1..p-2} (1-k*x) ); Gf(1, x) = 1. For the first right hand column p = 1, for the second p = 2, etc..
From Peter Bala, Jul 23 2013: (Start)
Conjectural explicit formula: T(n,k) = Stirling2(n,n-k) + (n-k)*Sum_{j = 0..k-1} (-1)^j*Stirling2(n, n+1+j-k)*j! for 0 <= k <= n.
The n-th row polynomial R(n,x) appears to satisfy the recurrence equation R(n,x) = n*x^(n-1) + Sum_{k = 1..n-1} binomial(n,k+1)*x^(n-k-1)*R(k,x). The row polynomials appear to have only real zeros. (End)

Edited by Johannes W. Meijer, Sep 23 2012

A163972 The MC polynomials.

Original entry on oeis.org

1, 0, 3, 1, 0, 2, 45, 22, 3, 0, 0, 10, 107, 61, 13, 1, 0, -48, 20, 2100, 14855, 9168, 2390, 300, 15, 0, 0, -336, 92, 6320, 33765, 21803, 6378, 1010, 85, 3, 0, 11520, -2016, -198296, 33012, 2199246, 9547461, 6331782, 1994265, 362474, 39375, 2394, 63

Offset: 1

Johannes W. Meijer, Aug 13 2009

The a(n,p) polynomials, see below with the extra p for the column number, generate the coefficients of the left hand columns of triangle A163940. These polynomials are interesting in their own right. They have many curious properties; e.g., for p >= 1: a(n=1, p) = p, a(n=0, p) = 0, a(n = -1, p) = (-1)^(p+1), a(n=-2,p) = (-1)^(p+1)*(2)^(p-2) and a(n = -(2*p+1), 2*p) = 0, which is the outermost zero of the a(n, 2*p); for p >= 10: a(n=-10, p) = -362880*10^(p-10); etc.
The numbers in the denominators of the a(n,p) are the Minkowski numbers A053657.
The Maple program generates the coefficients of the polynomials that appear in the numerators of the a(n,p), see the sequence above. We have made use of a nice little program that Peter Luschny recently wrote for the Minkowski numbers! For the an(p,k) in the Maple program for p >= 1 we have 0 <= k <= (2*p-2). A word of caution: The value of nmax has to be chosen sufficiently large in order to let Maple find the o.g.f.s.
The zero patterns of the a(n,p) polynomials resemble the Montezuma Cypress (Taxodium mucronatum). A famous Montezuma Cypress is 'El Arbol del Tule' (the Tule tree) in Mexico. It is the second stoutest tree in the world, circumference 36 meters, and is approximately 1500 years old. Considering this I propose to call the a(n,p) polynomials the MC polynomials.
The row sums equal n*A053657(n). [Johannes W. Meijer, Nov 29 2012]

			The a(n,p) formulas of the first few left hand columns of the A163940 triangle (p is the column number):
a(n,1) = (1)/1
a(n,2) = (0 + 3*n + n^2)/2
a(n,3) = (0 + 2*n + 45*n^2+ 22*n^3 + 3*n^4)/24
a(n,4) = (0 + 0*n + 10*n^2 + 107*n^3 + 61*n^4 + 13*n^5 + n^6)/48
a(n,5) = (0 - 48*n + 20*n^2 + 2100*n^3 + 14855*n^4 + 9168*n^5 + 2390*n^6 + 300*n^7 + 15*n^8)/5760
a(n,6) = (0 + 0*n -336*n^2 +92*n^3 +6320*n^4 +33765*n^5 +21803*n^6 +6378*n^7 +1010*n^8 +85*n^9 +3*n^10)/11520
a(n,7) = (0 + 11520*n -2016*n^2 -198296*n^3 +33012*n^4 +2199246*n^5 +9547461*n^6+ 6331782*n^7 +1994265*n^8 +362474*n^9 +39375*n^10 +2394*n^11 +63*n^12)/2903040

Johannes W. Meijer, The zeros of the MC polynomials, pdf and jpg.

A000012, A000096, A163943 and A163944 are the first four left hand columns of A163940.

Cf. A053657 (Minkowski), A163402 and A075264.

pmax:=6; nmax:=70; with(genfunc): A053657 := proc(n) local P, p, q, s, r; P := select(isprime, [$2..n]); r:=1; for p in P do s := 0: q := p-1; do if q > (n-1) then break fi; s := s + iquo(n-1, q); q := q*p; od; r := r * p^s; od; r end: for px from 1 to nmax do Gf(px):= convert(series(1/((1-(px-1)*x)^2*product((1-k*x), k=1..px-2)),x,nmax+1-px),polynom): for qy from 0 to nmax-px do a(px+qy,qy):=coeff(Gf(px),x,qy) od; od: for p from 1 to pmax do f(x):=0: for ny from p to nmax do f(x):=f(x)+a(ny,p-1)*x^(ny-p) od: f(x):= series(f(x),x, nmax): Gx:=convert(%, ratpoly): rgf_sequence('recur',Gx,x,G,n): a(n,p):=sort(simplify (rgf_expand(Gx,x,n)),n): f(p):=sort(a(n,p)*A053657(p),n,ascending): for k from 0 to 2*p-2 do an(p,k):= coeff(f(p),n,k) od; od: T:=1: for p from 1 to pmax do for k from 0 to 2*p-2 do a(T):=an(p,k): T:=T+1 od: od: seq(a(n),n=1..T-1); for p from 1 to pmax do seq(an(p,k),k=0..2*p-2) od; for p from 1 to pmax do MC(n,p):=sort(a(n,p),n,ascending) od;

A163943 Third left hand column of triangle A163940.

Original entry on oeis.org

0, 3, 17, 52, 121, 240, 428, 707, 1102, 1641, 2355, 3278, 4447, 5902, 7686, 9845, 12428, 15487, 19077, 23256, 28085, 33628, 39952, 47127, 55226, 64325, 74503, 85842, 98427, 112346, 127690, 144553, 163032, 183227, 205241, 229180, 255153

Offset: 0

Johannes W. Meijer, Aug 13 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A163972.
Equals the third left hand column of A163940.
A163944 is another left hand column.

LinearRecurrence[{5,-10,10,-5,1},{0,3,17,52,121},40] (* Harvey P. Dale, Feb 25 2017 *)

x='x+O('x^50); concat([0], Vec(x*(3 +2*x -3*x^2 +x^3)/(1-x)^5)) \\ G. C. Greubel, Aug 13 2017

G.f.: x*(3 + 2*x - 3*x^2 + x^3)/(1-x)^5.
a(n)= (2*n + 45*n^2 + 22*n^3 + 3*n^4)/24.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
E.g.f.: (1/24)*x*(72 + 132*x + 40*x^2 + 3*x^3)*exp(x). - G. C. Greubel, Aug 13 2017

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A163940 Triangle related to the divergent series 1^m1! - 2^m2! + 3^m3! - 4^m4! + ... for m >= -1.

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A163972 The MC polynomials.

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A163943 Third left hand column of triangle A163940.

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cp's OEIS Frontend

A163940 Triangle related to the divergent series 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... for m >= -1.

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A163972 The MC polynomials.

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A163943 Third left hand column of triangle A163940.

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A163940 Triangle related to the divergent series 1^m1! - 2^m2! + 3^m3! - 4^m4! + ... for m >= -1.