Real-rooted atlas

A038303 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111011
2120100121
313030010001331
414060040001000014641
515010001000050000100000161051
616015002000015000060000010000001771561
7170210035000350000210000070000001000000019487171
818028005600070000056000002800000080000000100000000214358881
91903600840001260000126000008400000036000000090000000010000000002357947691
101100450012000021000002520000021000000012000000004500000000100000000001000000000025937424601
111110550016500033000004620000046200000033000000001650000000055000000000110000000000100000000000285311670611
1211206600220000495000079200000924000000792000000049500000000220000000000660000000000120000000000010000000000003138428376721
1311307800286000715000012870000017160000001716000000012870000000071500000000028600000000007800000000000130000000000001000000000000034522712143931
1411409100364000100100002002000003003000000343200000003003000000002002000000000100100000000003640000000000091000000000000140000000000000100000000000000379749833583241
Technical details
Node id
n-e49156f63ff7a53f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d23628de2a16b138

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, 10;
  2: 1, 20,  100;
  3: 1, 30,  300,  1000;
  4: 1, 40,  600,  4000,   10000;
  5: 1, 50, 1000, 10000,   50000,   100000;
  6: 1, 60, 1500, 20000,  150000,   600000,  1000000;
  7: 1, 70, 2100, 35000,  350000,  2100000,  7000000,  10000000;
  8: 1, 80, 2800, 56000,  700000,  5600000, 28000000,  80000000, 100000000;
  9: 1, 90, 3600, 84000, 1260000, 12600000, 84000000, 360000000, 900000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (1+10*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[(1 + 10*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] = (1 + 10*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.