Real-rooted atlas

A038255 → basis A013611

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(7 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2495616121
3343588336641331
4240154884704179225614641
51680748020548803136089601024161051
61176494033685762404390401881604300840961771561
7823543329417256471525378240307328010536962007041638419487171
857648012635337652706752602362884302592019668992561971291750465536214358881
9403536072075328364743607686324810245421265923097866241180139522890137641287682621442357947691
1028247524916141442804150656720632481024063248102404337012736206524416067436544014450688018350080104857625937424601
111977326743124289109563551117416060876298560695729126405565833011231804760064129815347203709009920706478080807403524194304285311670611
12138412872019491168366429829386294456817878656073051558272066789996134444526664089621808978329677889208320197813862403391094784352321536167772163138428376721
139688901040771974693445224677037752645170426957696738632422528075973620602885788466331648330769504665614175835914244500265369601028632084481603062988815267266566710886434522712143931
1467822307284954257845827922015291416465646063803804928723859774077448272683132313670908712562688463077306531842315386532659288205201244162520148606976523667243008748096061446576668672268435456379749833583241

Generated b-files: A001020 (row sums), A013625 (reversed triangle), A038270 ?

Computed in your browser ?

Technical details
Node id
n-ade19c5257c8a726 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-d36c4f1a2e9bd6b7

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        7,         4;
  2:       49,        56,        16;
  3:      343,       588,       336,        64;
  4:     2401,      5488,      4704,      1792,       256;
  5:    16807,     48020,     54880,     31360,      8960,      1024;
  6:   117649,    403368,    576240,    439040,    188160,     43008,      4096;
  7:   823543,   3294172,   5647152,   5378240,   3073280,   1053696,    200704,    16384;
  8:  5764801,  26353376,  52706752,  60236288,  43025920,  19668992,   5619712,   917504,   65536;
  9: 40353607, 207532836, 474360768, 632481024, 542126592, 309786624, 118013952, 28901376, 4128768, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (7+4*x)*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[(7 + 4*x)*P[n - 1, 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] = (7 + 4*x)*v[i - 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)

Generated with polytool c48629374597.

A013625: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       7;
  2:     16,      56,       49;
  3:     64,     336,      588,       343;
  4:    256,    1792,     4704,      5488,      2401;
  5:   1024,    8960,    31360,     54880,     48020,     16807;
  6:   4096,   43008,   188160,    439040,    576240,    403368,    117649;
  7:  16384,  200704,  1053696,   3073280,   5378240,   5647152,   3294172,    823543;
  8:  65536,  917504,  5619712,  19668992,  43025920,  60236288,  52706752,  26353376,   5764801;
  9: 262144, 4128768, 28901376, 118013952, 309786624, 542126592, 632481024, 474360768, 207532836, 40353607;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+4*x)*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[(7 + 4*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (7 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038270: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        7,         4;
  2:       49,        56,        16;
  3:      343,       588,       336,        64;
  4:     2401,      5488,      4704,      1792,       256;
  5:    16807,     48020,     54880,     31360,      8960,      1024;
  6:   117649,    403368,    576240,    439040,    188160,     43008,      4096;
  7:   823543,   3294172,   5647152,   5378240,   3073280,   1053696,    200704,    16384;
  8:  5764801,  26353376,  52706752,  60236288,  43025920,  19668992,   5619712,   917504,   65536;
  9: 40353607, 207532836, 474360768, 632481024, 542126592, 309786624, 118013952, 28901376, 4128768, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (7+4*x)*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[(7 + 4*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (7 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-1b27bff9bafb5517
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-1b27bff9bafb5517
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-1b27bff9bafb5517

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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. [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
  4. [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