{"id":"n-f8babbbcd1463e71","revision":1,"first_index":0,"fingerprint":"f6-eca81e4e5309813a","name":"gamma_polynomial of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3"],["9","42"],["27","672"],["81","7854","9408"],["243","83076","379848"],["729","847770","9946608","5317872"],["2187","8538936","216567456","406133952"],["6561","85603686","4279372272","18843048000","5685875328"],["19683","856771692","79940038248","691877999232","703867479168"],["59049","8570196978","1442944276032","22200874658928","50878767821952","9854144708352"],["177147","85710236640","25489914754080","654028311000960","2820490783380480","1799062997932032"],["531441","857129647038","444128040636000","18194691109922880","133217196054862080","186825006823397376","25186881971048448"],["1594323","8571385752468","7670174841397320","486414927189381120","5658087130934246400","14559513238361376768","6364509879752306688"],["4782969","85714147691466","131712965709634512","12640400282561209200","223069363923520988160","949327602223129565184","897902500600031625216","89103138316532293632"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","51","699","17343","463167","16112979","631242531","28893905847","1476542308023","84385301721291","5299157717481339","363868761130045263","27076203922466055087","2172174804025637790339"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","-14","0"],["0","14","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","7","-28"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162693,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-c57a2725c54d20ba"]},{"id":162694,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-c57a2725c54d20ba"]},{"id":162695,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-c57a2725c54d20ba"]}],"region_exact":{},"incoming_edges":[{"id":"e-c57a2725c54d20ba","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-f8babbbcd1463e71","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":"c09804b831e41e5bbd6ca56a417814dc641e5ca32ee4c745ec8efce6bc7ae7ea","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-64714e0ad28cd8ed","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     3;\n  2:     9,        42;\n  3:    27,       672;\n  4:    81,      7854,        9408;\n  5:   243,     83076,      379848;\n  6:   729,    847770,     9946608,      5317872;\n  7:  2187,   8538936,   216567456,    406133952;\n  8:  6561,  85603686,  4279372272,  18843048000,   5685875328;\n  9: 19683, 856771692, 79940038248, 691877999232, 703867479168;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(14*n-14)*x)*P(n-1,x) + (7*x-28*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + (14*n - 14)*x)*P[n - 1, x] + (7*x - 28*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + (14*n - 14)*x)*v[i - 1] + (7*x - 28*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-64714e0ad28cd8ed","steps":["\u03b3"],"edges":["e-c57a2725c54d20ba"],"text":"A257627 \u2192 \u03b3"},"headline":"A257627 \u2192 \u03b3","chain":null}
