{"id":"n-7da7145fe72dc08d","revision":1,"first_index":1,"fingerprint":"f6-eae0d43706dd3f1f","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+8*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","6"],["1","16"],["1","38","32"],["1","84","248"],["1","178","1248","496"],["1","368","5168","6976"],["1","750","19184","58912","13952"],["1","1516","66552","389120","305408"],["1","3050","220880","2222000","3861760","610816"],["1","6120","711440","11538560","37084800","19111936"],["1","12262","2244480","56114400","300809600","337180416","38223872"],["1","24548","6978680","260369280","2177420800","4429559296","1616288768"],["1","49122","21476096","1167093360","14532273920","48351563776","37891377152","3232577536"],["1","98272","65607216","5097895360","91334863360","464496669696","652052150272","177426128896"],["1","196574","199376720","21834940192","548436433280","4065668105216","9209331748864","5331721932800","354852257792"]],"degrees":[1,1,2,2,3,3,4,4,5,5,6,6,7,7,8],"row_sums":["7","17","71","333","1923","12513","92799","762597","6918507","68452857","734585031","8490641373","105196410963","1390473413073","19532044991439"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","6"]],"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":162564,"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-b61cfbcaf1cd183b"]},{"id":162565,"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-b61cfbcaf1cd183b"]},{"id":162566,"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-b61cfbcaf1cd183b"]}],"region_exact":{},"incoming_edges":[{"id":"e-b61cfbcaf1cd183b","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-7da7145fe72dc08d","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":"01024de7f83fc6a4e7419c68497f9dd9c43f1776e6e716e13f0714fc633259c1","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-2255c682502f6ca3","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+8*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,    6;\n   2: 1,   16;\n   3: 1,   38,     32;\n   4: 1,   84,    248;\n   5: 1,  178,   1248,      496;\n   6: 1,  368,   5168,     6976;\n   7: 1,  750,  19184,    58912,    13952;\n   8: 1, 1516,  66552,   389120,   305408;\n   9: 1, 3050, 220880,  2222000,  3861760,   610816;\n  10: 1, 6120, 711440, 11538560, 37084800, 19111936;\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 + 6*x.","mathematica":"P[1, x_] := 1 + 6*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 + 6*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-2255c682502f6ca3","steps":["\u03b3"],"edges":["e-b61cfbcaf1cd183b"],"text":"A166340 \u2192 \u03b3"},"headline":"A166340 \u2192 \u03b3","chain":null}