Real-rooted atlas

A038324 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 20\,x + 20\,n\right)\,P_{n}(x) = \left(441 + 840\,x + 400\,x^{2} + 420\,n + 400\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1412061
2128116404003321
33572176860492008000169781
49353612857680307440013120001600008339441
52353220193536100142884000102480000328000003200000398432301
657585824128238641205612166000571536000030744000007872000006400000018652848361
7138083454818062015374019767048840026190108000020003760000086083200000183680000001280000000859768867621

All 15 rows

Technical details
Node id
n-7da3df1306d64907 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8ceec15cbd981f14

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:            41,             20;
  2:          1281,           1640,             400;
  3:         35721,          76860,           49200,            8000;
  4:        935361,        2857680,         3074400,         1312000,          160000;
  5:      23532201,       93536100,       142884000,       102480000,        32800000,         3200000;
  6:     575858241,     2823864120,      5612166000,      5715360000,      3074400000,       787200000,        64000000;
  7:   13808345481,    80620153740,    197670488400,    261901080000,    200037600000,     86083200000,     18368000000,     1280000000;
  8:  325997025921,  2209335276960,   6449612299200,  10542426048000,  10476043200000,   6401203200000,   2295552000000,   419840000000,   25600000000;
  9: 7602394731561, 58679464665780, 198840174926400, 386976737952000, 474409172160000, 377137555200000, 192036096000000, 59028480000000, 9446400000000, 512000000000;
%F Formula
The row polynomials satisfy (20*n+1+20*x)*P(n,x) = (420*n+441+(400*n+840)*x+400*x^2)*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[Together[((420*n + 441 + (400*n + 840)*x + 400*x^2)*P[n - 1, x])/(20*n + 1 + 20*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] = ((420*n + 441 + (400*n + 840)*x + 400*x^2)*v[i - 1])/(20*n + 1 + 20*x)); 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

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.