Real-rooted atlas

A013613 → basis A007318

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(7 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17613
2498436169
33438827562162197
424018232105846048129628561
51680772030123480105840453607776371293
611764960505212965401481760952560326592466564826809
782354349412581270609218151560155584808001504228614427993662748517
857648013953006411859019220329747221781872014936140864012032156764161679616815730721
940353607311299254106731172821346234562744515872235244217613442526724938071041058158081007769610604499373
10282475249242121642093389776202134623456032019351840329341904642352442176011522165760370355328070543872060466176137858491849
1119773267431864336643479900141860205457507640352212870240422655444288362276095104221801690880950578675202715939072046558955523627970561792160394037
121384128720114236752549667116119162419176034046403698235137520507186533145650718653314563726268406784199621521792076046294016019554761318430474952704217678233623298085122481
139688901040710796204016785552333494344174501909822243739326639048057692468145312659342493089285651507083622436331116966144173005318886405931610933248138661034803219808719257613060694016302875106592253
146782230728498138676874188453440568704761554653378416323664540106267046282068753600648076945540343687912109917071365934082437803523390904250173441453244678645764529593803571297062724362241279948013568783641640963937376385699289

Generated b-files: A001022 (row sums), A013627 (reversed triangle), A038272 ?

Computed in your browser ?

Technical details
Node id
n-bc97c3236a99a08c · revision 1 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-8e9037e0ccb64a62

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,         6;
  2:       49,        84,         36;
  3:      343,       882,        756,        216;
  4:     2401,      8232,      10584,       6048,       1296;
  5:    16807,     72030,     123480,     105840,      45360,       7776;
  6:   117649,    605052,    1296540,    1481760,     952560,     326592,      46656;
  7:   823543,   4941258,   12706092,   18151560,   15558480,    8001504,    2286144,    279936;
  8:  5764801,  39530064,  118590192,  203297472,  217818720,  149361408,   64012032,  15676416,   1679616;
  9: 40353607, 311299254, 1067311728, 2134623456, 2744515872, 2352442176, 1344252672, 493807104, 105815808, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (7+6*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 + 6*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 + 6*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.

A013627: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,         7;
  2:       36,        84,        49;
  3:      216,       756,       882,        343;
  4:     1296,      6048,     10584,       8232,       2401;
  5:     7776,     45360,    105840,     123480,      72030,      16807;
  6:    46656,    326592,    952560,    1481760,    1296540,     605052,     117649;
  7:   279936,   2286144,   8001504,   15558480,   18151560,   12706092,    4941258,     823543;
  8:  1679616,  15676416,  64012032,  149361408,  217818720,  203297472,  118590192,   39530064,   5764801;
  9: 10077696, 105815808, 493807104, 1344252672, 2352442176, 2744515872, 2134623456, 1067311728, 311299254, 40353607;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+6*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 + 6*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 + 6*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.

A038272: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        7,         6;
  2:       49,        84,         36;
  3:      343,       882,        756,        216;
  4:     2401,      8232,      10584,       6048,       1296;
  5:    16807,     72030,     123480,     105840,      45360,       7776;
  6:   117649,    605052,    1296540,    1481760,     952560,     326592,      46656;
  7:   823543,   4941258,   12706092,   18151560,   15558480,    8001504,    2286144,    279936;
  8:  5764801,  39530064,  118590192,  203297472,  217818720,  149361408,   64012032,  15676416,   1679616;
  9: 40353607, 311299254, 1067311728, 2134623456, 2744515872, 2352442176, 1344252672, 493807104, 105815808, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (7+6*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 + 6*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 + 6*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-c08496e978bf378e
  • 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-c08496e978bf378e
  • 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-c08496e978bf378e

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 (3)

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

References

  1. [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