Real-rooted atlas

A088996 → basis A007318 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-x + n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
225310
3625341580
424142311298105880
512092628334289320794512320
672068762718456897664214094710395209440
7504057492279676751671120460811501026050761351354188800
84032053553630997881020850820911929272699362209213810157220202702596342400
9362880550526437002204144572968361751997600932217662418154466981326190915965344594252504902400
103628800619473604746220562148771556636440636212882610113180418936631725575818610781804844397199758565472907572642169600
113991680075762576065217371763360355029211513267265827536124484546899564266656869310696948097077519027010987515990613969110137493105752324549427200
12479001600100095955209568623859255324255855221544737286125952386409274119612563434271761121706494618851590762567143036463312787299894757977224911301643031623414322581359229952000
13622702080014209978176014941609896969584309225976418392224357561312428665358623043059926964015279772304299736853511065370216572371713707894524897901356512117358381630156033886927267579058535806253091650738176000
148717829120021575265062402475494796326417452267588545684452546320193629670591372674807803841294217888156079430288224492385030071312756527705800275690613240802606889104171518152606794478765616118002188551739831420490975213458046676875126757680265216000
Technical details
Node id
n-00cc6ee2ee07a006 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d445f8c61d9cb983

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 0)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1,       1;
  2:      2,       5,        3;
  3:      6,      25,       34,        15;
  4:     24,     142,      311,       298,       105;
  5:    120,     926,     2833,      4289,      3207,       945;
  6:    720,    6876,    27184,     56897,     66421,     40947,     10395;
  7:   5040,   57492,   279676,    751671,   1204608,   1150102,    605076,    135135;
  8:  40320,  535536,  3099788,  10208508,  20911929,  27269936,  22092138,  10157220,   2027025;
  9: 362880, 5505264, 37002204, 144572968, 361751997, 600932217, 662418154, 466981326, 190915965, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (n+(2*n-1)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + (2*n - 1)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + (2*n - 1)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.