{"id":"n-dbf55b6b36baa8f7","revision":0,"first_index":1,"fingerprint":"f6-4748f217d058d591","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["8","16"],["20","96","96"],["40","336","768","512"],["70","896","3456","5120","2560"],["112","2016","11520","28160","30720","12288"],["168","4032","31680","112640","199680","172032","57344"],["240","7392","76032","366080","931840","1290240","917504","262144"],["330","12672","164736","1025024","3494400","6881280","7798784","4718592","1179648"],["440","20592","329472","2562560","11182080","29245440","46792704","44826624","23592960","5242880"],["572","32032","617760","5857280","31682560","105283584","222265344","298844160","247726080","115343360","23068672"],["728","48048","1098240","12446720","81469440","333398016","889061376","1568931840","1816657920","1326448640","553648128","100663296"],["910","69888","1867008","24893440","193489920","952565760","3111714816","6903300096","10445783040","10611589120","6920601600","2617245696","436207616"],["1120","99008","3055104","47297536","429977600","2500485120","9779675136","26462650368","50139758592","66322432000","59978547200","35332816896","12213813248","1879048192"],["1360","137088","4837248","85995520","902952960","6112296960","28116566016","90729086976","208915660800","344876646400","404855193600","329772957696","177100292096","56371445760","8053063680"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","24","212","1656","12102","84816","577576","3851472","25275466","163795752","1050721404","6683872392","42219328910","265089657120","1655897134160"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["-1","-2","-1"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["2","8","10","4"],["1","4","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","6","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179567,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-947702eb73f76d5c"]},{"id":179568,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-947702eb73f76d5c"]},{"id":179569,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-947702eb73f76d5c"]}],"region_exact":{},"incoming_edges":[{"id":"e-947702eb73f76d5c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-dbf55b6b36baa8f7","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":"0e1ea3f6e504b21d449eca33a93b25188f04194b855ca1ce7ca684a9339ae229","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-92be85f47bd42a26","role":"sequence","name":"Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-7091d188d4739a26","kind":"fingerprint_related","node_a_id":"n-8c995547482f9480","node_b_id":"n-dbf55b6b36baa8f7","fingerprint":"f6-4748f217d058d591","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4748f217d058d591"},"other_id":"n-8c995547482f9480","other":{"id":"n-8c995547482f9480","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","first_index":1,"fingerprint":"f6-4748f217d058d591","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["16","8"],["96","96","20"]],"card_rows":[["2"],["16","8"],["96","96","20"],["512","768","336","40"]],"row_sums_preview":["2","24","212","1656","12102","84816","577576","3851472"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-92be85f47bd42a26","steps":["x \u21a6 x+1"],"edges":["e-ccf60822171bc8ef"],"text":"A375853 \u2192 x \u21a6 x+1"},"headline":"A375853 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   2;\n   2:   8,    16;\n   3:  20,    96,     96;\n   4:  40,   336,    768,     512;\n   5:  70,   896,   3456,    5120,     2560;\n   6: 112,  2016,  11520,   28160,    30720,    12288;\n   7: 168,  4032,  31680,  112640,   199680,   172032,    57344;\n   8: 240,  7392,  76032,  366080,   931840,  1290240,   917504,   262144;\n   9: 330, 12672, 164736, 1025024,  3494400,  6881280,  7798784,  4718592,  1179648;\n  10: 440, 20592, 329472, 2562560, 11182080, 29245440, 46792704, 44826624, 23592960, 5242880;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (n+2+(4*n+8)*x+(5*n+10)*x^2+(2*n+4)*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.","mathematica":"P[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (4*n + 8)*x + (5*n + 10)*x^2 + (2*n + 4)*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((n + 2 + (4*n + 8)*x + (5*n + 10)*x^2 + (2*n + 4)*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-92be85f47bd42a26","steps":["x \u21a6 x/(x+1)"],"edges":["e-947702eb73f76d5c"],"text":"A375853 \u2192 x \u21a6 x/(x+1)"},"headline":"A375853 \u2192 x \u21a6 x/(x+1)","chain":null}
