Real-rooted atlas

A013611 → D → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
240128168
3300192030725292
42000192006144065536148176
512500160000768000163840013107203889620
6750001200000768000024576000393216002516582498018424
74375008400000672000002867200006881280008808038404697620482401451388
8250000056000000537600000286720000091750400001761607680018790481920858993459257634833312
9140625003600000004032000000258048000001032192000002642411520004227858432003865470566401546188226561361622936996
107812500022500000002880000000021504000000010321920000003303014400000704643072000096636764160007730941132800274877906944031771201863240
114296875001375000000019800000000016896000000009461760000000363331584000009688842240000017716740096000021260088115200015118284881920048378511622144733914763040844
12234375000082500000000132000000000012672000000000811008000000003633315840000001162661068800000265751101440000042520176230400004535485464576000290271069732864084442493013196816813320026026608
1312695312500487500000000858000000000091520000000000658944000000000337379328000000012595494912000000345476431872000006909528637440000098268851732480000943380976631808005488762045857792014636698788954112382503030592105332
146835937500028437500000005460000000000064064000000000051251200000000002952069120000000012595494912000000040305583718400000096733400924160000017197049053184000002201222278807552000192106671605022720010245689152267878402522015791327477768650453153390689816
Technical details
Node id
n-b451fc07991937e5 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-833b492313a93e6e

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:        4;
  2:       40,       128;
  3:      300,      1920,       3072;
  4:     2000,     19200,      61440,       65536;
  5:    12500,    160000,     768000,     1638400,      1310720;
  6:    75000,   1200000,    7680000,    24576000,     39321600,     25165824;
  7:   437500,   8400000,   67200000,   286720000,    688128000,    880803840,    469762048;
  8:  2500000,  56000000,  537600000,  2867200000,   9175040000,  17616076800,  18790481920,   8589934592;
  9: 14062500, 360000000, 4032000000, 25804800000, 103219200000, 264241152000, 422785843200, 386547056640, 154618822656;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+16*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 16*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 16*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.