Real-rooted atlas

A038271 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(12 + 5\,n x\right)\,P_{n}(x) = \left(144 + 120\,x + 25\,x^{2} + 60\,n x + 25\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1121022
214424075459
31728432027005009248
4207366912064800240003125181781
524883210368001296000720000187500187503507882
629859841492992023328000172800006750000135000010937566733279
7358318082090188803919104003628800001890000005670000091875006250001255153588
84299816962866544640627056640069672960004536000000181440000044100000060000000351562523389304361
951597803523869835264096745881600125411328000979776000004898880000015876000000324000000037968750019531250432496961342
106191736422451597803520014511882240002149908480000195955200000011757312000004762800000001296000000002278125000023437500001074218757945387725299
11743008370688681091006464021284093952000354734899200003695155200000025866086400000125737920000004276800000000100237500000015468750000014179687500585937500145151560832328
1289161004482568916100448256030649095290880056757583872000066512793600000053210234880000030177100800000012317184000000036085500000000742500000000010209375000008437500000031738281252638936015687741
1310699320537907211590930582732804346598968524800885418308403200011528884224000000103759958016000006725182464000000320246784000000011258676000000002895750000000005308875000000065812500000004951171875001708984375047775023452840402
141283918464548864149790487530700806085238555934720013522752346521600019368525496320000019368525496320000014122883174400000076859228160000000315242928000000009729720000000000222972750000000036855000000000041589843750000287109375000091552734375861698288862816519
Technical details
Node id
n-1e15874f42e02f6f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-601043bbbbe9602c

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:         12,          10;
  2:        144,         240,          75;
  3:       1728,        4320,        2700,          500;
  4:      20736,       69120,       64800,        24000,        3125;
  5:     248832,     1036800,     1296000,       720000,      187500,       18750;
  6:    2985984,    14929920,    23328000,     17280000,     6750000,     1350000,      109375;
  7:   35831808,   209018880,   391910400,    362880000,   189000000,    56700000,     9187500,     625000;
  8:  429981696,  2866544640,  6270566400,   6967296000,  4536000000,  1814400000,   441000000,   60000000,   3515625;
  9: 5159780352, 38698352640, 96745881600, 125411328000, 97977600000, 48988800000, 15876000000, 3240000000, 379687500, 19531250;
%F Formula
The row polynomials satisfy (12+5*n*x)*P(n,x) = (144+(60*n+120)*x+(25*n+25)*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[((144 + (60*n + 120)*x + (25*n + 25)*x^2)*P[n - 1, x])/(12 + 5*n*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] = ((144 + (60*n + 120)*x + (25*n + 25)*x^2)*v[i - 1])/(12 + 5*n*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

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.