{"id":"n-8272e45f524cffe5","revision":0,"first_index":0,"fingerprint":"f6-44af577288c301d1","name":"mobius_x_over_x_plus_one of Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","12","12"],["1","94","220","128"],["1","824","3432","4480","1872"],["1","7386","54504","124288","114768","37600"],["1","66436","963796","3464832","5294512","3717824","990784"],["1","597878","19444452","102311168","228525872","257240032","144512576","32333824"],["1","5380848","434693904","3199817984","9824291808","15812723456","14083318528","6597042176","1272660224"],["1","48427570","10318501264","105798528256","434258986976","943222606912","1187865671424","875206156288","351359873280","59527313920"],["1","435848060","252248078556","3723549752064","20162434708384","57003389151104","95308460940416","98213780070400","61535235140864","21569850059776","3252626013184"],["1","3922632462","6247798324140","140643518931840","993457017275040","3559730186839872","7586794172440704","10250393525886976","8908933484678400","4845751275458048","1505444531203072","204354574172160"],["1","35303692072","155606563234552","5705235272566400","51948707126343600","231321636498700032","611489966432987904","1040657291869200384","1176978759228092160","883438968717322240","424246456087402496","118291143545618432","14594815769038848"],["1","317733228554","3884303667980280","246266043998267008","2868687603575686320","15682144083037365792","50495005990823286528","105415031707213455360","149105278163470016256","144896471148759969280","95649922452851726336","41097735454888034304","10389246780979793920","1174376539738169344"],["1","2859599056884","97049764251910692","11142678641300571520","166555669218807705552","1111604015627322467392","4307044531075272293568","10825529780704076980224","18637769312159062506240","22523849249013407693824","19167593509049540115456","11282970352858267090944","4383363048320590999552","1012922132309620998144","105595092426069327872"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","25","443","10609","338547","14498185","784965803","51229928929","3967606065891","361022009762809","38001754007842715","4558828622414199121","617024050590736979283","93536036424027189717865"],"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":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["1","12","37","50","32","8"],["0","-8","-36","-60","-44","-12"],["0","0","4","12","12","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","48","112","128","72","16"],["0","0","-8","-32","-48","-32","-8"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","4","20","40","40","20","4"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188540,"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-89361ce7cf75bc63"]},{"id":188541,"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-89361ce7cf75bc63"]},{"id":188542,"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-89361ce7cf75bc63"]}],"region_exact":{},"incoming_edges":[{"id":"e-89361ce7cf75bc63","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-8272e45f524cffe5","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":"834dd7b810d423eddcdda3fa9049c448c3fb140b033de42041531b25236dd7c9","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-65e47d0258ed4170","role":"sequence","name":"Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).","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,        2;\n  2: 1,       12,          12;\n  3: 1,       94,         220,          128;\n  4: 1,      824,        3432,         4480,         1872;\n  5: 1,     7386,       54504,       124288,       114768,        37600;\n  6: 1,    66436,      963796,      3464832,      5294512,      3717824,        990784;\n  7: 1,   597878,    19444452,    102311168,    228525872,    257240032,     144512576,     32333824;\n  8: 1,  5380848,   434693904,   3199817984,   9824291808,  15812723456,   14083318528,   6597042176,   1272660224;\n  9: 1, 48427570, 10318501264, 105798528256, 434258986976, 943222606912, 1187865671424, 875206156288, 351359873280, 59527313920;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(8*n-12)*x+(4*n^2-36*n+37)*x^2+(12*n^2-60*n+50)*x^3+(12*n^2-44*n+32)*x^4+(4*n^2-12*n+8)*x^5)*P(n-1,x) + (8*x-(8*n-48)*x^2-(32*n-112)*x^3-(48*n-128)*x^4-(32*n-72)*x^5-(8*n-16)*x^6)*d/dx P(n-1,x) + (4*x^2+20*x^3+40*x^4+40*x^5+20*x^6+4*x^7)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((1 - (8*n - 12)*x + (4*n^2 - 36*n + 37)*x^2 + (12*n^2 - 60*n + 50)*x^3 + (12*n^2 - 44*n + 32)*x^4 + (4*n^2 - 12*n + 8)*x^5)*P[n - 1, x] + (8*x - (8*n - 48)*x^2 - (32*n - 112)*x^3 - (48*n - 128)*x^4 - (32*n - 72)*x^5 - (8*n - 16)*x^6)*D[P[n - 1, x], x] + (4*x^2 + 20*x^3 + 40*x^4 + 40*x^5 + 20*x^6 + 4*x^7)*D[P[n - 1, x], {x, 2}])/(1 + 2*x + x^2)]];\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 - (8*n - 12)*x + (4*n^2 - 36*n + 37)*x^2 + (12*n^2 - 60*n + 50)*x^3 + (12*n^2 - 44*n + 32)*x^4 + (4*n^2 - 12*n + 8)*x^5)*v[i - 1] + (8*x - (8*n - 48)*x^2 - (32*n - 112)*x^3 - (48*n - 128)*x^4 - (32*n - 72)*x^5 - (8*n - 16)*x^6)*deriv(v[i - 1], x) + (4*x^2 + 20*x^3 + 40*x^4 + 40*x^5 + 20*x^6 + 4*x^7)*deriv(deriv(v[i - 1], x), x))/(1 + 2*x + x^2)); 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-65e47d0258ed4170","steps":["x \u21a6 x/(x+1)"],"edges":["e-89361ce7cf75bc63"],"text":"A160562 \u2192 x \u21a6 x/(x+1)"},"headline":"A160562 \u2192 x \u21a6 x/(x+1)","chain":null}