{"id":"n-e4f7b836c7f2dd88","revision":0,"first_index":0,"fingerprint":"f6-f145e7ba99e5a3dd","name":"mobius_x_over_x_plus_one of Stirling2 triangle with scaled diagonals (powers of 6).","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":false,"rows":[["36","90","55"],["216","900","1188","505"],["1296","8424","18396","16764","5497"],["7776","79056","257904","379476","262674","69823"],["46656","769824","3539376","7467984","8171136","4526766","1007407"],["279936","7884864","48942144","138440016","212464944","183791448","84587208","16157905"],["1679616","84820608","687849408","2500341696","5044124016","6055976016","4323079944","1699808760","284214097"],["10077696","948983040","9840870144","44752295232","114059433312","178313753808","175573925568","106598730456","36562378590","5432922775"],["60466176","10914144768","143377060608","802163886336","2516191205760","4922271942048","6270400674000","5226557109696","2758190810508","838347151938","112034017735"],["362797056","127765029888","2129679184896","14504634696960","54966851195904","130929950658816","207229848494976","222344195809968","160475957992008","74825005993596","20405048801004","2476196276617"],["2176782336","1511775332352","32305324317696","265906559462400","1200396261862656","3412620395108352","6525975837219840","8653031205595776","8007411171968352","5090850105880824","2124591270696324","525094036854516","58332035387017"],["13060694016","17999813136384","501215016775680","4957461958136832","26362278172408320","88088643561265920","199408862618284032","317697272134585344","361872666052696512","294089816515995072","166951710291592368","63023988635968140","14236103981084010","1457666574447247"],["78364164096","215070448361472","7958862592782336","94125994661007360","584276985530265600","2267411052751971840","5982634129483954944","11220623034137616384","15296016274627291776","15255103845432044736","11043956108756018592","5660205389693850384","1949748632069946696","405391719241083798","38485034941511935"],["470184984576","2574811339702272","129273364936802304","1820428659474518016","13096105678126338048","58470449671066798080","177668026409726650368","386236538142579332352","617362801853607233280","734664372510061339776","651152736499708109952","424771785371504685024","198336743124473167200","62802365750448160752","12092615379942333840","1069787864231083297"],["2821109907456","30858710722707456","2144066647094968320","35846875143582105600","297239700883417214976","1515501625941817602048","5252893792017644648448","13076778560072806914048","24129754412112130809600","33566224943856766466304","35413452243880764953472","28228302386875329158016","16749501351921393309216","7179417204714396261792","2102846707737008770896","376943553255904443888","31213730550761076769"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["181","2809","50377","1056709","25529149","692548465","20681894161","672093370621","23600508469573","890415496931689","35898028156468441","1538665698387069877","69806152212731871949","3339676605561411239137","167958092017142651318305"],"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":[["36","90","55"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["6","7","-5","-6"],["0","-6","-12","-6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","18","18","6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191744,"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-5ce01ff9bf619d62"]},{"id":191745,"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-5ce01ff9bf619d62"]},{"id":191746,"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-5ce01ff9bf619d62"]}],"region_exact":{},"incoming_edges":[{"id":"e-5ce01ff9bf619d62","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-e4f7b836c7f2dd88","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":"34db7166f45feecb3596bfa91c1365ab0a3922ab8cf96b73a37899f3cf2d26e4","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-adb5eb667a229c4c","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 6).","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:        36,           90,            55;\n  1:       216,          900,          1188,            505;\n  2:      1296,         8424,         18396,          16764,           5497;\n  3:      7776,        79056,        257904,         379476,         262674,           69823;\n  4:     46656,       769824,       3539376,        7467984,        8171136,         4526766,         1007407;\n  5:    279936,      7884864,      48942144,      138440016,      212464944,       183791448,        84587208,        16157905;\n  6:   1679616,     84820608,     687849408,     2500341696,     5044124016,      6055976016,      4323079944,      1699808760,       284214097;\n  7:  10077696,    948983040,    9840870144,    44752295232,   114059433312,    178313753808,    175573925568,    106598730456,     36562378590,     5432922775;\n  8:  60466176,  10914144768,  143377060608,   802163886336,  2516191205760,   4922271942048,   6270400674000,   5226557109696,   2758190810508,   838347151938,   112034017735;\n  9: 362797056, 127765029888, 2129679184896, 14504634696960, 54966851195904, 130929950658816, 207229848494976, 222344195809968, 160475957992008, 74825005993596, 20405048801004, 2476196276617;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (6-(6*n-7)*x-(12*n+5)*x^2-(6*n+6)*x^3)*P(n-1,x) + (6*x+18*x^2+18*x^3+6*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 36 + 90*x + 55*x^2.","mathematica":"P[0, x_] := 36 + 90*x + 55*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6 - (6*n - 7)*x - (12*n + 5)*x^2 - (6*n + 6)*x^3)*P[n - 1, x] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*D[P[n - 1, x], x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 36 + 90*x + 55*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6 - (6*n - 7)*x - (12*n + 5)*x^2 - (6*n + 6)*x^3)*v[i - 1] + (6*x + 18*x^2 + 18*x^3 + 6*x^4)*deriv(v[i - 1], x))/(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-adb5eb667a229c4c","steps":["x \u21a6 x/(x+1)"],"edges":["e-5ce01ff9bf619d62"],"text":"A075501 \u2192 x \u21a6 x/(x+1)"},"headline":"A075501 \u2192 x \u21a6 x/(x+1)","chain":null}