Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.

OEIS matches ?

• A009966: 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,           240,            100;
  1:         1728,          4320,           3600,           1000;
  2:        20736,         69120,          86400,          48000,           10000;
  3:       248832,       1036800,        1728000,        1440000,          600000,          100000;
  4:      2985984,      14929920,       31104000,       34560000,        21600000,         7200000,         1000000;
  5:     35831808,     209018880,      522547200,      725760000,       604800000,       302400000,        84000000,       10000000;
  6:    429981696,    2866544640,     8360755200,    13934592000,     14515200000,      9676800000,      4032000000,      960000000,      100000000;
  7:   5159780352,   38698352640,   128994508800,   250822656000,    313528320000,    261273600000,    145152000000,    51840000000,    10800000000,    1000000000;
  8:  61917364224,  515978035200,  1934917632000,  4299816960000,   6270566400000,   6270566400000,   4354560000000,  2073600000000,   648000000000,  120000000000,   10000000000;
  9: 743008370688, 6810910064640, 28378791936000, 70946979840000, 118244966400000, 137952460800000, 114960384000000, 68428800000000, 28512000000000, 7920000000000, 1320000000000, 100000000000;

The row polynomials satisfy P(n,x) = (12+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 240*x + 100*x^2.

P[0, x_] := 144 + 240*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 10*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 + 240*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 10*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-2f1e093ae220e593
• 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-2f1e093ae220e593, e-3f0a5f7ceded39cb
• 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-2f1e093ae220e593, e-3f0a5f7ceded39cb
• 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-2f1e093ae220e593
• 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-3f0a5f7ceded39cb
• 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-8872d945c5d0b661
• 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-8872d945c5d0b661

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-8872d945c5d0b661 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-3f0a5f7ceded39cb root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-2f1e093ae220e593 root: theorem check

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

Used in 7 proof steps

• e-022a6c0efe3a8d25 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• e-0c8cc5263f914382 basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• e-344e4e749f34f1c6 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• e-79186619a2157f5c basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• e-865f1fe93a5cf324 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• e-0e5e881cd5b1385e obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• e-0e5e881cd5b1385e obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.

Related sequences ?

Same fingerprint (6)

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

• A038334 → basis A013610
• A013612 → basis A007318
• A038220 → basis A013610
• A038271 → basis A013609
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^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