{"id":"n-a19d1a6e6c32556d","revision":1,"first_index":0,"fingerprint":"f6-37e18a3c0000c110","name":"gamma_polynomial of Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.","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"],["1"],["1","16"],["1","176"],["1","1632","2816"],["1","14752","100096"],["1","132848","2409728","1601536"],["1","1195728","49467648","117149696"],["1","10761664","936608256","5303189504","1874395136"],["1","96855104","16955460096","192522665984","231557103616"],["1","871696080","299090593280","6169503457280","16882472386560","3704913657856"],["1","7845264880","5196117184000","182950283386880","951969810022400","692140576342016"],["1","70607384096","89463710270720","5155722209280000","46051026401689600","74072314511097856","11074249221472256"],["1","635466457056","1532180256057600","140344410146652160","2014633203346636800","5984230584663146496","2912952276207271936"],["1","5719198113712","26158906449421056","3729244210538598400","82201469646864056320","406524110238919950336","429977729597987356672","46607236419316350976"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","17","177","4449","114849","4144113","167813073","8124954561","441132084801","27056851791057","1832264632200177","136442846661194529","11053693290086222049","969065954739273847473"],"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":[["1","-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":162684,"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-15b23c3dbd64ff2a"]},{"id":162685,"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-15b23c3dbd64ff2a"]},{"id":162686,"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-15b23c3dbd64ff2a"]}],"region_exact":{},"incoming_edges":[{"id":"e-15b23c3dbd64ff2a","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-a19d1a6e6c32556d","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":"1e5adfc6e815f3561a18072362e705cec32d099790fb126c1f2018279a555f22","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-90af30f849fc554d","role":"sequence","name":"Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.","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: 1;\n  2: 1,       16;\n  3: 1,      176;\n  4: 1,     1632,        2816;\n  5: 1,    14752,      100096;\n  6: 1,   132848,     2409728,      1601536;\n  7: 1,  1195728,    49467648,    117149696;\n  8: 1, 10761664,   936608256,   5303189504,   1874395136;\n  9: 1, 96855104, 16955460096, 192522665984, 231557103616;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(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[(1 + (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] = (1 + (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-90af30f849fc554d","steps":["\u03b3"],"edges":["e-15b23c3dbd64ff2a"],"text":"A167884 \u2192 \u03b3"},"headline":"A167884 \u2192 \u03b3","chain":null}