{"id":"n-bb7184253a86be79","revision":0,"first_index":0,"fingerprint":"f6-9daba0cbde89f7fd","name":"mobius_x_over_x_plus_one of shift_x_plus_one of Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","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"],["3","5"],["10","33","27"],["37","182","296","159"],["151","987","2401","2574","1025"],["674","5499","17822","28661","22853","7221"],["3263","31938","129436","277868","333059","211195","55307"],["17007","194291","945842","2542424","4073462","3888155","2046062","457631"],["94828","1239137","7047167","22775086","45731431","58399102","46296406","20821820","4065569"],["562595","8279468","53894906","203615635","491882561","787688723","835878016","566597582","222528839","38566021"],["3535027","57874781","424503349","1836587500","5189043539","10001581399","13314651739","12085080884","7155186808","2494551957","388757083"],["23430840","422516115","3449044634","16820682061","54443418290","122771813555","196778451550","224118242471","177711012431","93408443532","29283189524","4146851583"],["163254885","3215822672","28923095532","157031246920","573095687366","1480884429962","2777621977278","3809557386812","3791001489812","2668868200673","1261428600127","359313295050","46636281185"],["1192059223","25472533865","250326882132","1497797238596","6086827569792","17738467816846","38132018264788","61211330367006","73360799404111","64820253490412","41035060791886","17621724972075","4599985276361","551163837685"],["9097183602","209627964005","2235246164191","14616433789540","65473528934870","212501857281888","515268577803060","948097231971728","1330051356464297","1415452453720801","1124663525154321","646864565631968","254547431185161","61332093027247","6825500514059"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","70","674","7138","82730","1042066","14164874","206470546","3209494346","52951354066","923377096586","16957740768274","326932420504778","6598148526790738"],"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","1","-5","-9","-4"],["2","10","18","14","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-7","-19","-25","-16","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181966,"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-348ac5b40d437979"]},{"id":181967,"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-348ac5b40d437979"]},{"id":181968,"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-348ac5b40d437979"]}],"region_exact":{},"incoming_edges":[{"id":"e-348ac5b40d437979","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-bb7184253a86be79","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":"21fc09afa69b62701115483fe7c7ce01b2be7e4a6b5e3a7a8529364ac25305f4","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-f339d17ab0e2d2a9","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","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:      3,       5;\n  2:     10,      33,       27;\n  3:     37,     182,      296,       159;\n  4:    151,     987,     2401,      2574,      1025;\n  5:    674,    5499,    17822,     28661,     22853,      7221;\n  6:   3263,   31938,   129436,    277868,    333059,    211195,     55307;\n  7:  17007,  194291,   945842,   2542424,   4073462,   3888155,   2046062,    457631;\n  8:  94828, 1239137,  7047167,  22775086,  45731431,  58399102,  46296406,  20821820,   4065569;\n  9: 562595, 8279468, 53894906, 203615635, 491882561, 787688723, 835878016, 566597582, 222528839, 38566021;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2*n+1+(10*n+1)*x+(18*n-5)*x^2+(14*n-9)*x^3+(4*n-4)*x^4)*P(n-1,x) - (1+7*x+19*x^2+25*x^3+16*x^4+4*x^5)*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[Together[((2*n + 1 + (10*n + 1)*x + (18*n - 5)*x^2 + (14*n - 9)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 4*x^5)*D[P[n - 1, x], x])/(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] = ((2*n + 1 + (10*n + 1)*x + (18*n - 5)*x^2 + (14*n - 9)*x^3 + (4*n - 4)*x^4)*v[i - 1] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 4*x^5)*deriv(v[i - 1], 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-0990b5be9f344750","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-2b15edbdb8f7710e","e-348ac5b40d437979"],"text":"A126351 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A126351 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}