Real-rooted atlas

A038303 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
220222
3300603363
44000120012045324
550000200003000200573205
66000003000006000060003006966306
770000004200000105000014000010500420712400927
88000000056000000168000002800000280000168005608155897368
99000000007200000002520000005040000063000005040002520072091929229929
10100000000009000000000360000000084000000012600000012600000840000360009001023579476910
111100000000001100000000004950000000013200000000231000000027720000023100000132000049500110011285311670611
1212000000000001320000000000660000000000198000000000396000000005544000000554400000396000001980000660001320123423740047332
131300000000000015600000000000858000000000028600000000006435000000001029600000001201200000010296000006435000028600008580015601340799568897373
14140000000000000182000000000000109200000000000400400000000001001000000000018018000000002402400000002402400000018018000001001000004004000109200182014483317970015034
Technical details
Node id
n-769b27b46bada28a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d23628de2a16b138

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         0;
  1:         1;
  2:        20,         2;
  3:       300,        60,         3;
  4:      4000,      1200,       120,        4;
  5:     50000,     20000,      3000,      200,       5;
  6:    600000,    300000,     60000,     6000,     300,      6;
  7:   7000000,   4200000,   1050000,   140000,   10500,    420,     7;
  8:  80000000,  56000000,  16800000,  2800000,  280000,  16800,   560,   8;
  9: 900000000, 720000000, 252000000, 50400000, 6300000, 504000, 25200, 720, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (10*n+n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((10*n + n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((10*n + n*x)*v[i - 1])/(n - 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.