Real-rooted atlas

A013622 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(32 + 80\,x + 50\,x^{2} + 40\,n + 50\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1181028
2224360100684
3243267205400100015552
424576972801344007200010000338256
52375681228800243200022400009000001000007138368
6222822414254080368640004864000033600000108000001000000147386304
720447232155975680498892800860160000851200000470400000126000000100000002993075712
8184549376163577856062390272001330380800017203200000136192000006272000000144000000010000000059997563136
916441671681660944384073610035200187170816000299335680000309657600000204288000000806400000001620000000010000000001190155742208
10144955146241644167168008304721920002453667840000467927040000059867136000005160960000000291840000000010080000000001800000000001000000000023406396263424
111267015352321594506608640904291942400030450647040000674758656000001029439488000001097564160000008110080000000040128000000000123200000000001980000000000100000000000457019805007872
121099511627776152041842278409567039651840036171677696000091351941120000016194207744000002058878976000000188153856000000012165120000000005350400000000001478400000000002160000000000010000000000008869040590934016
13948328778956814293651161088098827197480960041457171824640001175579525120000023751504691200000350874501120000003823632384000000030575001600000000175718400000000006955520000000000174720000000000023400000000000010000000000000171211044451074048
148136386045542413276602905395201000555581276160046119358824448000145100101386240000329162267033600000554201776128000000701749002240000000669135667200000000475611136000000000246005760000000000885248000000000002038400000000000025200000000000001000000000000003290028448776044544
Technical details
Node id
n-09e3e314b59270f4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d37b0b195ff0bdee

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:         18,          10;
  2:        224,         360,         100;
  3:       2432,        6720,        5400,         1000;
  4:      24576,       97280,      134400,        72000,        10000;
  5:     237568,     1228800,     2432000,      2240000,       900000,       100000;
  6:    2228224,    14254080,    36864000,     48640000,     33600000,     10800000,      1000000;
  7:   20447232,   155975680,   498892800,    860160000,    851200000,    470400000,    126000000,    10000000;
  8:  184549376,  1635778560,  6239027200,  13303808000,  17203200000,  13619200000,   6272000000,  1440000000,   100000000;
  9: 1644167168, 16609443840, 73610035200, 187170816000, 299335680000, 309657600000, 204288000000, 80640000000, 16200000000, 1000000000;
%F Formula
The row polynomials satisfy (5*n-1+5*x)*P(n,x) = (40*n+32+(50*n+80)*x+50*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[((40*n + 32 + (50*n + 80)*x + 50*x^2)*P[n - 1, x])/(5*n - 1 + 5*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] = ((40*n + 32 + (50*n + 80)*x + 50*x^2)*v[i - 1])/(5*n - 1 + 5*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.