Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^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:          100,           240,           144;
  1:         1000,          3600,          4320,           1728;
  2:        10000,         48000,         86400,          69120,          20736;
  3:       100000,        600000,       1440000,        1728000,        1036800,          248832;
  4:      1000000,       7200000,      21600000,       34560000,       31104000,        14929920,         2985984;
  5:     10000000,      84000000,     302400000,      604800000,      725760000,       522547200,       209018880,        35831808;
  6:    100000000,     960000000,    4032000000,     9676800000,    14515200000,     13934592000,      8360755200,      2866544640,      429981696;
  7:   1000000000,   10800000000,   51840000000,   145152000000,   261273600000,    313528320000,    250822656000,    128994508800,    38698352640,     5159780352;
  8:  10000000000,  120000000000,  648000000000,  2073600000000,  4354560000000,   6270566400000,   6270566400000,   4299816960000,  1934917632000,   515978035200,   61917364224;
  9: 100000000000, 1320000000000, 7920000000000, 28512000000000, 68428800000000, 114960384000000, 137952460800000, 118244966400000, 70946979840000, 28378791936000, 6810910064640, 743008370688;

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

P[0, x_] := 100 + 240*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(10 + 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] = 100 + 240*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (10 + 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, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-50ef4b08e4862a18
• 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-08ecf9f1e7a97c44, e-50ef4b08e4862a18
• 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-08ecf9f1e7a97c44, e-50ef4b08e4862a18
• 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-50ef4b08e4862a18
• 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-08ecf9f1e7a97c44
• 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-0e5da81d270807c6
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 10] to roots in (-inf, 0]: an input root r <= 10 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-0e5da81d270807c6

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-0e5da81d270807c6 root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-08ecf9f1e7a97c44 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-50ef4b08e4862a18 root: theorem check

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

Used in 7 proof steps

• e-047b920dffe4a1a4 basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• e-0be1d4b95988d4e0 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• e-61350e73fec19600 basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• e-6524eda715a9c2e7 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• e-f9a8e29d679c9a8f basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• e-4dc01c187e0a54c1 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• e-4dc01c187e0a54c1 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.

Related sequences ?

Same fingerprint (6)

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

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