{"id":"n-ba66df50c3e69feb","revision":1,"first_index":1,"fingerprint":"f6-050c3ba74718fd87","name":"gamma_polynomial of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","4"],["1","12"],["1","30","24"],["1","68","192"],["1","146","984","384"],["1","304","4120","5472"],["1","622","15400","46608","10944"],["1","1260","53664","309632","241152"],["1","2538","178632","1775168","3063552","482304"],["1","5096","576504","9244256","29519104","15148032"],["1","10214","1821240","45048080","240038080","268002816","30296064"],["1","20452","5668000","209332160","1740767360","3528321536","1284083712"],["1","40930","17453944","939352640","11634487040","38577602816","30158515200","2568167424"],["1","81888","53344152","4106489440","73202077440","371079461376","519730428928","141179400192"],["1","163806","162161544","17599529104","439927197120","3251305852416","7348907616256","4247817775104","282358800384"]],"degrees":[1,1,2,2,3,3,4,4,5,5,6,6,7,7,8],"row_sums":["5","13","55","261","1515","9897","73575","605709","5502195","54492993","585216495","6768193221","83895619995","1109351283417","15588079095735"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0","0"],["0","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-4"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162573,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-e41185e16509dc0b"]},{"id":162574,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-e41185e16509dc0b"]},{"id":162575,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-e41185e16509dc0b"]}],"region_exact":{},"incoming_edges":[{"id":"e-e41185e16509dc0b","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-ba66df50c3e69feb","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"d4d027207187711e83a89c27ed56aa3581671c0668e492f08b47cc0b1c17c8d4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9a3eef8528def021","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1,    4;\n   2: 1,   12;\n   3: 1,   30,     24;\n   4: 1,   68,    192;\n   5: 1,  146,    984,     384;\n   6: 1,  304,   4120,    5472;\n   7: 1,  622,  15400,   46608,    10944;\n   8: 1, 1260,  53664,  309632,   241152;\n   9: 1, 2538, 178632, 1775168,  3063552,   482304;\n  10: 1, 5096, 576504, 9244256, 29519104, 15148032;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*n*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 4*x.","mathematica":"P[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*n*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 4*x; for(n = 2, N, my(i = n); v[i] = (1 + 2*n*x)*v[i - 1] + (x - 4*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9a3eef8528def021","steps":["\u03b3"],"edges":["e-e41185e16509dc0b"],"text":"A166344 \u2192 \u03b3"},"headline":"A166344 \u2192 \u03b3","chain":null}