{"id":"n-03d4f0c316134406","revision":1,"first_index":0,"fingerprint":"f6-bb44a1e208d0448a","name":"gamma_polynomial of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 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","20"],["8","220"],["16","1780","2200"],["32","13100","62000"],["64","93300","1137000","620000"],["128","656940","17376000","33280000"],["256","4607540","241359000","1087040000","332800000"],["512","32273260","3172760400","28134040000","29062400000"],["1024","225958900","40332253000","636916700000","1483394000000","290624000000"],["2048","1581814700","502063748000","13247519080000","58111336000000","37514728000000"],["4096","11072928180","6167128299000","260352286720000","1940825346000000","2756237736000000","375147280000000"],["8192","77510988780","75112832406000","4919359138160000","58319294815200000","152051432712000000","67129467680000000"],["16384","542577986420","909880197637800","90389260265260000","1627379625605600000","7021353423984000000","6709685947120000000","671294676800000000"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","24","228","3996","75132","1850364","51313068","1665806796","60401474172","2451492912924","109377228644748","5338740849951276","282494744688762972","16121013356550500604"],"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","-10","0"],["0","10","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5","-20"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162651,"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-9458cf74e817fa3d"]},{"id":162652,"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-9458cf74e817fa3d"]},{"id":162653,"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-9458cf74e817fa3d"]}],"region_exact":{},"incoming_edges":[{"id":"e-9458cf74e817fa3d","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-03d4f0c316134406","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":"85cf21eb563d533ce03ff942ec44f7a442bef148d0df8c5698181ac3222b46f6","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-ee6acb764200c6c7","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 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":[],"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,       20;\n  3:   8,      220;\n  4:  16,     1780,       2200;\n  5:  32,    13100,      62000;\n  6:  64,    93300,    1137000,      620000;\n  7: 128,   656940,   17376000,    33280000;\n  8: 256,  4607540,  241359000,  1087040000,   332800000;\n  9: 512, 32273260, 3172760400, 28134040000, 29062400000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(10*n-10)*x)*P(n-1,x) + (5*x-20*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 + (10*n - 10)*x)*P[n - 1, x] + (5*x - 20*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 + (10*n - 10)*x)*v[i - 1] + (5*x - 20*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-ee6acb764200c6c7","steps":["\u03b3"],"edges":["e-9458cf74e817fa3d"],"text":"A257614 \u2192 \u03b3"},"headline":"A257614 \u2192 \u03b3","chain":null}