A038338 → Pₙ + Pₙ₊₁

OEIS matches ?

• A168702: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168846: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168894: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168798: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A167940: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168990: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168750: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A168942: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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.

Triangle begins:
  0:          1872,            5472,            5328,             1728;
  1:         22464,           88128,          129600,            84672,            20736;
  2:        269568,         1327104,         2612736,          2571264,          1264896,           248832;
  3:       3234816,        19160064,        47278080,         62208000,         46033920,         18164736,          2985984;
  4:      38817792,       268738560,       797257728,       1313832960,       1298903040,        770383872,        253808640,         35831808;
  5:     465813504,      3690676224,     12791955456,      25333088256,      31352832000,      24831442944,      12290310144,       3475685376,        429981696;
  6:    5589762048,     49877876736,    197791580160,     457500524544,     680231043072,     674211299328,     445461037056,     189191946240,      46868004864,       5159780352;
  7:   67077144576,    665611665408,   2972033482752,    7863505256448,   13652778811392,   16253308108800,   13436068036608,    7615835799552,    2832719413248,     624333422592,     61917364224;
  8:  804925734912,   8792265719808,  43651741777920,  130026464870400,  258195408814080,  358873043042304,  356272513744896,  252622846033920,  125382662553600,   41484634030080,   8235009441792,    743008370688;
  9: 9659108818944, 115166297456640, 629328089972736, 2084138479779840, 4658662484213760, 7404821422276608, 8581746681446400, 7306744317345792, 4536066103050240, 2002407559004160, 596635721662464, 107736213749760, 8916100448256;

The row polynomials satisfy P(n,x) = (12+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 5472*x + 5328*x^2 + 1728*x^3.

P[0, x_] := 1872 + 5472*x + 5328*x^2 + 1728*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1872 + 5472*x + 5328*x^2 + 1728*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 12*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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-328ff92d3afa5881

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

• obreschkoff_combination e-328ff92d3afa5881

  Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check liu_wang e-e4b4d500ff341474 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-969ef99525bc6c6a root: theorem check

      Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
2. [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
3. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.