{"id":"n-a71082a9c4388af6","revision":1,"first_index":0,"fingerprint":"f6-a9bc49a1eb5b44eb","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) = 8*x + 2.","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"],["2"],["4","32"],["8","448"],["16","4864","7168"],["32","49664","284672"],["64","499200","7507968","4554752"],["128","4998144","167092224","358678528"],["256","49995776","3407511552","17346068480","5738856448"],["512","499990528","66134802432","669078519808","750195310592"],["1024","4999979008","1246425382912","22686825709568","57622409510912","12003124969472"],["2048","49999953920","23075654205440","709514305208320","3411118768783360","2348048353067008"],["4096","499999899648","422561769062400","21031845206425600","172739182555299840","262351731730415616","37568773649072128"],["8192","4999999782912","7686111827066880","600915881807052800","7892189346697052160","22074080416216645632","10273694097826906112"],["16384","49999999533056","139230012848996352","16730613030081003520","335637016550089687040","1558486525216863289344","1573240564869744295936","164379105565230497792"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","36","456","12048","334368","12561984","530769024","26542432512","1485908623872","93563785552896","6491807081220096","494114594910179328","40848570854374514688","3648613105244857319424"],"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":[["2","-16","0"],["0","16","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-32"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162690,"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-9580ca662e76a47a"]},{"id":162691,"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-9580ca662e76a47a"]},{"id":162692,"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-9580ca662e76a47a"]}],"region_exact":{},"incoming_edges":[{"id":"e-9580ca662e76a47a","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-a71082a9c4388af6","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":"27634120304e276d037612db1bd91353355ca5cf3ead89a09eb1d703a327abf4","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-fb16da45c5e69d3d","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) = 8*x + 2.","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":[{"id":"l-a3e6e371b0dc578f","kind":"fingerprint_related","node_a_id":"n-ada634934a2d63b9","node_b_id":"n-a71082a9c4388af6","fingerprint":"f6-a9bc49a1eb5b44eb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a9bc49a1eb5b44eb"},"other_id":"n-ada634934a2d63b9","other":{"id":"n-ada634934a2d63b9","name":"gamma_polynomial of Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).","first_index":1,"fingerprint":"f6-a9bc49a1eb5b44eb","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","8"]],"card_rows":[["1"],["1"],["1","8"],["1","56"]],"row_sums_preview":["1","1","9","57","753","10449","196281","4146633"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d023ddeb9da11712","steps":["\u03b3"],"edges":["e-e113043a8d357c9d"],"text":"A142459 \u2192 \u03b3"},"headline":"A142459 \u2192 \u03b3","chain":null}}],"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:   2;\n  2:   4,        32;\n  3:   8,       448;\n  4:  16,      4864,        7168;\n  5:  32,     49664,      284672;\n  6:  64,    499200,     7507968,      4554752;\n  7: 128,   4998144,   167092224,    358678528;\n  8: 256,  49995776,  3407511552,  17346068480,   5738856448;\n  9: 512, 499990528, 66134802432, 669078519808, 750195310592;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(16*n-16)*x)*P(n-1,x) + (8*x-32*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[(2 + (16*n - 16)*x)*P[n - 1, x] + (8*x - 32*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] = (2 + (16*n - 16)*x)*v[i - 1] + (8*x - 32*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-fb16da45c5e69d3d","steps":["\u03b3"],"edges":["e-9580ca662e76a47a"],"text":"A257618 \u2192 \u03b3"},"headline":"A257618 \u2192 \u03b3","chain":null}