{"id":"n-22136b111878a0ba","revision":1,"first_index":0,"fingerprint":"f6-3b6b2c52864dfad0","name":"gamma_polynomial of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.","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"],["5"],["25","20"],["125","340"],["625","3880","1360"],["3125","37160","43280"],["15625","322620","835440","173120"],["78125","2633340","12680880","8587840"],["390625","20620880","166794720","246636800","34351360"],["1953125","156846160","1996053600","5381720320","2419662080"],["9765625","1168235620","22356174880","99119995520","96036250880","9678648320"],["48828125","8568274340","238589113760","1626868147840","2834391189760","913469731840"],["244140625","62126357880","2455759900080","24576044811520","69384448423680","47714740254720","3653878927360"],["1220703125","446603255160","24586893415920","348920809729280","1491822904984320","1825872278599680","443833863802880"],["6103515625","3189699348620","240932583970320","4723257069995200","29163480437216000","57224542278681600","29455643027845120","1775335455211520"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","5","45","465","5865","83565","1346805","23980185","468794385","9956235285","228369070845","5621935285665","147847242815865","4135484574490365","122586386655784005"],"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":[["5","-4","0"],["0","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-8"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162633,"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-7978a139e0a4e7e6"]},{"id":162634,"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-7978a139e0a4e7e6"]},{"id":162635,"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-7978a139e0a4e7e6"]}],"region_exact":{},"incoming_edges":[{"id":"e-7978a139e0a4e7e6","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-22136b111878a0ba","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":"49684ade5798084ff7ea617b3752b898cc0738cae7bfcccb05311f26e5b5d0ed","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-3f0a1785078e99be","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.","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:       5;\n  2:      25,        20;\n  3:     125,       340;\n  4:     625,      3880,       1360;\n  5:    3125,     37160,      43280;\n  6:   15625,    322620,     835440,     173120;\n  7:   78125,   2633340,   12680880,    8587840;\n  8:  390625,  20620880,  166794720,  246636800,   34351360;\n  9: 1953125, 156846160, 1996053600, 5381720320, 2419662080;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+(4*n-4)*x)*P(n-1,x) + (2*x-8*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[(5 + (4*n - 4)*x)*P[n - 1, x] + (2*x - 8*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] = (5 + (4*n - 4)*x)*v[i - 1] + (2*x - 8*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-3f0a1785078e99be","steps":["\u03b3"],"edges":["e-7978a139e0a4e7e6"],"text":"A257615 \u2192 \u03b3"},"headline":"A257615 \u2192 \u03b3","chain":null}
