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

OEIS matches ?

• A009968: 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,           288,            144;
  1:         1728,          5184,           5184,            1728;
  2:        20736,         82944,         124416,           82944,           20736;
  3:       248832,       1244160,        2488320,         2488320,         1244160,          248832;
  4:      2985984,      17915904,       44789760,        59719680,        44789760,        17915904,         2985984;
  5:     35831808,     250822656,      752467968,      1254113280,      1254113280,       752467968,       250822656,        35831808;
  6:    429981696,    3439853568,    12039487488,     24078974976,     30098718720,     24078974976,     12039487488,      3439853568,       429981696;
  7:   5159780352,   46438023168,   185752092672,    433421549568,    650132324352,    650132324352,    433421549568,    185752092672,     46438023168,     5159780352;
  8:  61917364224,  619173642240,  2786281390080,   7430083706880,  13002646487040,  15603175784448,  13002646487040,   7430083706880,   2786281390080,   619173642240,   61917364224;
  9: 743008370688, 8173092077568, 40865460387840, 122596381163520, 245192762327040, 343269867257856, 343269867257856, 245192762327040, 122596381163520, 40865460387840, 8173092077568, 743008370688;

The row polynomials satisfy P(n,x) = (12+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 288*x + 144*x^2.

P[0, x_] := 144 + 288*x + 144*x^2;
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] = 144 + 288*x + 144*x^2; 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, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-e4b4d500ff341474
• 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-969ef99525bc6c6a, e-e4b4d500ff341474
• 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-969ef99525bc6c6a, e-e4b4d500ff341474
• 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-e4b4d500ff341474
• 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-969ef99525bc6c6a
• 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-d2f9900eddbe015a
• 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-d2f9900eddbe015a

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-d2f9900eddbe015a root: theorem check

  Geometric basis map (elementary) polytool ?

• parent_provers periodic_product e-969ef99525bc6c6a root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-e4b4d500ff341474 root: theorem check

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

Used in 8 proof steps

• e-44f842c6e5783c9b basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• e-85e8843cd1dca083 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• e-9ec880266e74c89d basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• e-a2ff444bbbdc7b83 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• e-c462762e7768ae01 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• e-5535504a50c93560 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• e-328ff92d3afa5881 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
• e-328ff92d3afa5881 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.

Related sequences ?

Same fingerprint (59)

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

• A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
• A007318 → ×(1+2n+n²) → D
• A356546 → basis A013609
• Riordan array (1-x,x/(1-x)).
• basis-map coefficients of A112227 → x ↦ x+1
• A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
• A344565 → basis A013609
• A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
• A135838 → basis A013609
• basis-map coefficients of A026729 → basis A007318
• A038208 → D
• A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
• A028326 → D
• A178821 → basis A013609
• A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
• A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
• A085881 → basis A013609
• A038338 → basis A013609
• A196347 → basis A013609
• A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
• A135065: A127733 * A007318 as infinite lower triangular matrices.
• A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
• A178822 → D
• A007318 → D → D
• A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
• A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
• A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
• A120910 → basis A013610
• basis-map coefficients of A026729, shifted → basis A007318
• A178822 → basis A013609
• A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A00010…
• A216973: Exponential Riordan array [x*exp(x),x].
• A178820 → D
• A007318: Binomial
• A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
• A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
• A344565 → D
• A003506: Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.
• A038330 → basis A013611
• A178821 → D
• A007318 → ×(−(1/2)n+(1/2)n²) → D
• A085880 → basis A013609
• A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
• A028326 → basis A013609
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
• basis-map coefficients of A021012 → x ↦ x+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → D
• A178820 → basis A013609
• basis-map coefficients of A113214, shifted → D
• A156992 → basis A013609
• basis-map coefficients of A113214, shifted → basis A013609
• A135838 → D
• A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
• A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
• A038325 → basis A013609
• A010054 → reverse → x ↦ x+1
• A085880 → D
• A038221 → basis A013609
• A027465 → 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