{"id":"n-9411e3ffcfcf94af","revision":0,"first_index":0,"fingerprint":"f6-0ddb4ba4487ba978","name":"mobius_x_over_x_plus_one of basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["9","30","25"],["18","96","170","100"],["36","264","724","880","400"],["72","672","2504","4656","4320","1600"],["144","1632","7696","19328","27264","20480","6400"],["288","3840","21920","69440","131840","150016","94720","25600"],["576","8832","59200","226560","541440","827392","789504","430080","102400"],["1152","19968","153728","689920","1989120","3820544","4888576","4018176","1925120","409600"],["2304","44544","387328","1994752","6737920","15597568","25059328","27590656","19922944","8519680","1638400"],["4608","98304","952832","5538816","21454848","58146816","112508928","155418624","150208512","96731136","37355520","6553600"],["9216","215040","2298880","14888960","65064960","202113024","457605120","760872960","922091520","794296320","461635584","162529280","26214400"],["18432","466944","5457920","38973440","189685760","664485888","1723662336","3352166400","4887674880","5276958720","4100456448","2171600896","702545920","104857600"],["36864","1007616","12783616","99778560","535265280","2087714816","6105268224","13598982144","23184015360","30104616960","29308747776","20745027584","10091495424","3019898880","419430400"],["73728","2162688","29597696","250691584","1469644800","6316490752","20561395712","51619037184","100763959296","152945295360","179035963392","158725046272","103163101184","46405779456","12918456320","1677721600"],["147456","4620288","67846144","619773952","3942055936","18511560704","66388754432","185483657216","408004067328","708946427904","969853108224","1033593946112","841226387456","505463963648","211460030464","55029268480","6710886400"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["64","384","2304","13824","82944","497664","2985984","17915904","107495424","644972544","3869835264","23219011584","139314069504","835884417024","5015306502144"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["9","30","25"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179289,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-bd4d56eecbcb1b74"]},{"id":179290,"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-bd4d56eecbcb1b74"]},{"id":179291,"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-bd4d56eecbcb1b74"]}],"region_exact":{},"incoming_edges":[{"id":"e-bd4d56eecbcb1b74","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-9411e3ffcfcf94af","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"d286c85e8bfa9fccb8749d637a6138db7c15bcbb86d683672573ecbcb935627b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2fe6a26f73f75da6","role":"sequence","name":"basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for 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:    9,    30,     25;\n  1:   18,    96,    170,     100;\n  2:   36,   264,    724,     880,      400;\n  3:   72,   672,   2504,    4656,     4320,     1600;\n  4:  144,  1632,   7696,   19328,    27264,    20480,      6400;\n  5:  288,  3840,  21920,   69440,   131840,   150016,     94720,     25600;\n  6:  576,  8832,  59200,  226560,   541440,   827392,    789504,    430080,    102400;\n  7: 1152, 19968, 153728,  689920,  1989120,  3820544,   4888576,   4018176,   1925120,   409600;\n  8: 2304, 44544, 387328, 1994752,  6737920, 15597568,  25059328,  27590656,  19922944,  8519680,  1638400;\n  9: 4608, 98304, 952832, 5538816, 21454848, 58146816, 112508928, 155418624, 150208512, 96731136, 37355520, 6553600;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2+6*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 30*x + 25*x^2.","mathematica":"P[0, x_] := 9 + 30*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + 6*x + 4*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 9 + 30*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 6*x + 4*x^2)*v[i - 1])/(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-f48bed27dbc29c5f","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-8866dbf598c953fc","e-bd4d56eecbcb1b74"],"text":"A391533 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A391533 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}