Real-rooted atlas

A253283: Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

A253283 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence from A253283

\(\left(1 - n\right)\,P_{n}(x) = \left(6 + 6\,x - 3\,n - 4\,n x\right)\,P_{n-1}(x) + \left(-9 - 14\,x + 3\,n + 4\,n x\right)\,P_{n-2}(x) + \left(4 - n\right)\,P_{n-3}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = x\)
\(P_{3}(x) = 2\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30235
403121025
504306035129
60560210280126681
706105560126012604623653
8071681260420069305544171619825

All 15 rows b-files: A253283 ?

Technical details
Node id
n-b0d6ec97a8d172b8 · revision 0 · JSON
Definition source
oeis_catalog:A253283
Fingerprint
f6-2c1fea4a558d4e6c

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

  • A253283 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A002002

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 2,   3;
   4: 0, 3,  12,   10;
   5: 0, 4,  30,   60,    35;
   6: 0, 5,  60,  210,   280,   126;
   7: 0, 6, 105,  560,  1260,  1260,    462;
   8: 0, 7, 168, 1260,  4200,  6930,   5544,   1716;
   9: 0, 8, 252, 2520, 11550, 27720,  36036,  24024,   6435;
  10: 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = -(3*n-6+(4*n-6)*x)*P(n-1,x) + (3*n-9+(4*n-14)*x)*P(n-2,x) - (n-4)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = x, P(3,x) = 2*x + 3*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := x;
P[3, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 6 + (4*n - 6)*x)*P[n - 1, x] + (3*n - 9 + (4*n - 14)*x)*P[n - 2, x] - (n - 4)*P[n - 3, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; v[3] = 2*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-(3*n - 6 + (4*n - 6)*x)*v[i - 1] + (3*n - 9 + (4*n - 14)*x)*v[i - 2] - (n - 4)*v[i - 3])/(-n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A253283 triangle, offset 0, from P1
%e Example
Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 2,   3;
  3: 0, 3,  12,   10;
  4: 0, 4,  30,   60,    35;
  5: 0, 5,  60,  210,   280,   126;
  6: 0, 6, 105,  560,  1260,  1260,    462;
  7: 0, 7, 168, 1260,  4200,  6930,   5544,   1716;
  8: 0, 8, 252, 2520, 11550, 27720,  36036,  24024,   6435;
  9: 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310;
%F Formula
The row polynomials satisfy -n*P(n,x) = -(3*n-3+(4*n-2)*x)*P(n-1,x) + (3*n-6+(4*n-10)*x)*P(n-2,x) - (n-3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 2*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[2, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (4*n - 2)*x)*P[n - 1, x] + (3*n - 6 + (4*n - 10)*x)*P[n - 2, x] - (n - 3)*P[n - 3, x])/(-n)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; v[3] = 2*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n - 3 + (4*n - 2)*x)*v[i - 1] + (3*n - 6 + (4*n - 10)*x)*v[i - 2] - (n - 3)*v[i - 3])/(-n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Used in 12

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Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.