Real-rooted atlas

A147716 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(15\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
260868
31350360241734
4270001080014406439304
5506250270000540004800160835210
69112500607500016200002160001440038417038284
715946875012757500042525000756000075600040320896337925966
827337500002551500000102060000022680000030240000241920010752020486565418768
94613203125049207500000229635000006123600000102060000010886400072576002764804608125563633938
1076886718750092264062500049207500000015309000000030618000000408240000036288000020736000691200102402371757529940
1112686308593750169150781250001014904687500036085500000008419950000001347192000001496880000011404800005702400016896002252844351865809878
1220759414062500030447140625000020298093750000081192375000000216513000000004041576000000538876800000513216000003421440000152064000405504049152822525511383192
1333734047851562505397447656250000395812828125000017591681250000005277504375000001125867600000001751349600000020015424000001667952000009884160000395366400958464010649615148178167973786
145449346191406250094455333984375000755642671875000003694253062500000012314176875000000295540245000000052540488000000070053984000000700539840000051891840000027675648000100638720022364160229376277328184921366236
Technical details
Node id
n-a25ec7b85bb2189d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-eb454baeb618da42

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:           2;
  2:          60,           8;
  3:        1350,         360,          24;
  4:       27000,       10800,        1440,         64;
  5:      506250,      270000,       54000,       4800,        160;
  6:     9112500,     6075000,     1620000,     216000,      14400,       384;
  7:   159468750,   127575000,    42525000,    7560000,     756000,     40320,     896;
  8:  2733750000,  2551500000,  1020600000,  226800000,   30240000,   2419200,  107520,   2048;
  9: 46132031250, 49207500000, 22963500000, 6123600000, 1020600000, 108864000, 7257600, 276480, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (15*n+2*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n + 2*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((15*n + 2*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.