Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.

OEIS matches ?

• A001023: row sums, 15 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:          144,            48,             4;
  1:         1728,           864,           144,            8;
  2:        20736,         13824,          3456,          384,           16;
  3:       248832,        207360,         69120,        11520,          960,          32;
  4:      2985984,       2985984,       1244160,       276480,        34560,        2304,         64;
  5:     35831808,      41803776,      20901888,      5806080,       967680,       96768,       5376,       128;
  6:    429981696,     573308928,     334430208,    111476736,     23224320,     3096576,     258048,     12288,      256;
  7:   5159780352,    7739670528,    5159780352,   2006581248,    501645312,    83607552,    9289728,    663552,    27648,     512;
  8:  61917364224,  103195607040,   77396705280,  34398535680,  10032906240,  2006581248,  278691840,  26542080,  1658880,   61440,   1024;
  9: 743008370688, 1362182012928, 1135151677440, 567575838720, 189191946240, 44144787456, 7357464576, 875888640, 72990720, 4055040, 135168, 2048;

The row polynomials satisfy P(n,x) = (12+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 48*x + 4*x^2.

P[0, x_] := 144 + 48*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 48*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 2*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, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-ca869591900be157
• 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-48411a1ad7cd0ada, e-ca869591900be157
• 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-48411a1ad7cd0ada, e-ca869591900be157
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-ca869591900be157
• 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-48411a1ad7cd0ada
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-33e9d74370728897
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 12] to roots in (-inf, 0]: an input root r <= 12 gives the root (r - a)/d <= 0; a larger root gives a positive root
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-33e9d74370728897

Proof chains

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

• basis_map_theorem e-33e9d74370728897 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-48411a1ad7cd0ada root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-ca869591900be157 root: theorem check

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

Used in 7 proof steps

• e-0eb3975e7d8d74fc basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
• e-6073a5a63ce56264 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
• e-81c568f66e8f893e basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
• e-b35ead6daa50b714 basis_map (A013610) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• e-ccf7b0982517b62a basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
• e-9d869e2e35cfa66c obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
• e-9d869e2e35cfa66c obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.

Related sequences ?

Same fingerprint (2)

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

• A013613: Triangle of coefficients in expansion of (1+6x)^n.
• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).

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