Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.

OEIS matches ?

• A001029: 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,           168,             49;
  1:         1728,          3024,           1764,            343;
  2:        20736,         48384,          42336,          16464,           2401;
  3:       248832,        725760,         846720,         493920,         144060,          16807;
  4:      2985984,      10450944,       15240960,       11854080,        5186160,        1210104,         117649;
  5:     35831808,     146313216,      256048128,      248935680,      145212480,       50824368,        9882516,        823543;
  6:    429981696,    2006581248,     4096770048,     4779565056,     3485099520,     1626379776,      474360768,      79060128,       5764801;
  7:   5159780352,   27088846848,    63207309312,    86032171008,    75278149632,    43912253952,    17076987648,    4269246912,     622598508,     40353607;
  8:  61917364224,  361184624640,   948109639680,  1474837217280,  1505562992640,  1053894094848,   512309629440,  170769876480,   37355910480,   4842432840,   282475249;
  9: 743008370688, 4767637045248, 13905608048640, 24334814085120, 28390616432640, 23185670086656, 13524974217216, 5635405923840, 1643660061120, 319600567440, 37286732868, 1977326743;

The row polynomials satisfy P(n,x) = (12+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 168*x + 49*x^2.

P[0, x_] := 144 + 168*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 7*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 + 168*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 7*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-de9af033a7745ebd
• 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-400776ae4466efdf, e-de9af033a7745ebd
• 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-400776ae4466efdf, e-de9af033a7745ebd
• 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-de9af033a7745ebd
• 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-400776ae4466efdf
• 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-159e2ac1fa72d770
• 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-159e2ac1fa72d770

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-159e2ac1fa72d770 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-400776ae4466efdf root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-de9af033a7745ebd root: theorem check

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

Used in 7 proof steps

• e-065a527a7a0e90c1 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• e-0d83dd77e9f6bf78 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• e-20f4338b8f3f0a8d basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• e-ab827b425fb4aadf basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• e-f4ff934f738cae69 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• e-946cc0d28b50f639 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
• e-946cc0d28b50f639 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.

Related sequences ?

Same fingerprint (3)

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

• A038284 → basis A013611
• A013613 → basis A013609
• A038335 → basis A013611

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