{"id":"n-2b38b764c2be7262","revision":0,"first_index":0,"fingerprint":"f6-3024ad5e34af04cb","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(k,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"],["2","5"],["3","18","24"],["5","45","129","116"],["8","98","435","824","560"],["13","200","1201","3505","4952","2704"],["21","390","2958","11700","25392","28608","13056"],["34","739","6771","33842","99473","171660","160816","63040"],["55","1370","14719","88976","330565","771920","1105120","886016","304384"],["89","2499","30801","218505","982236","2898573","5609808","6860256","4806912","1469696"],["144","4500","62586","509760","2690577","9608166","23491080","38798784","41402880","25763328","7096320"],["233","8020","124238","1142610","6927582","29054247","85955361","179266692","258145440","244320128","136725760","34264064"],["377","14172","241980","2480252","16987743","81864666","284466567","717774984","1304573328","1664874752","1415530752","719702016","165441536"],["610","24865","463938","5244052","40046335","218078235","870862155","2580901737","5673973608","9135282160","10465937920","8076998400","3762472960","798822400"],["987","43362","877746","10846644","91379466","554987964","2504915964","8531916840","22028895072","42890017920","61965264384","64397604864","45496541184","19554729984","3857055744"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","45","295","1925","12575","82125","536375","3503125","22879375","149428125","975934375","6373953125","41629109375","271885078125"],"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":[["3","6","3"],["3","6","3"]]},"terms":[{"coeff":{"coeffs":[["23","110","183","128","32"],["-11","-56","-111","-98","-32"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","43","131","185","124","32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177804,"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-160f8666c89b309f"]},{"id":177805,"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-160f8666c89b309f"]},{"id":177806,"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-160f8666c89b309f"]}],"region_exact":{},"incoming_edges":[{"id":"e-160f8666c89b309f","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-2b38b764c2be7262","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":"589eb47930eec64e3b1ee4d39414420edf3dafa5dfecd498153231944d4429fa","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-226ddb8fc3536584","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(k,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:  2,    5;\n  2:  3,   18,    24;\n  3:  5,   45,   129,    116;\n  4:  8,   98,   435,    824,    560;\n  5: 13,  200,  1201,   3505,   4952,    2704;\n  6: 21,  390,  2958,  11700,  25392,   28608,   13056;\n  7: 34,  739,  6771,  33842,  99473,  171660,  160816,   63040;\n  8: 55, 1370, 14719,  88976, 330565,  771920, 1105120,  886016,  304384;\n  9: 89, 2499, 30801, 218505, 982236, 2898573, 5609808, 6860256, 4806912, 1469696;\n","recurrence_sentence":"The row polynomials satisfy (3*n+3+(6*n+6)*x+(3*n+3)*x^2)*P(n,x) = -(11*n-23+(56*n-110)*x+(111*n-183)*x^2+(98*n-128)*x^3+(32*n-32)*x^4)*P(n-1,x) + (5+43*x+131*x^2+185*x^3+124*x^4+32*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[(-(11*n - 23 + (56*n - 110)*x + (111*n - 183)*x^2 + (98*n - 128)*x^3 + (32*n - 32)*x^4)*P[n - 1, x] + (5 + 43*x + 131*x^2 + 185*x^3 + 124*x^4 + 32*x^5)*D[P[n - 1, x], x])/(3*n + 3 + (6*n + 6)*x + (3*n + 3)*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] = (-(11*n - 23 + (56*n - 110)*x + (111*n - 183)*x^2 + (98*n - 128)*x^3 + (32*n - 32)*x^4)*v[i - 1] + (5 + 43*x + 131*x^2 + 185*x^3 + 124*x^4 + 32*x^5)*deriv(v[i - 1], x))/(3*n + 3 + (6*n + 6)*x + (3*n + 3)*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-21854f4257effb01","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-b1c4a0832f5f9447","e-160f8666c89b309f"],"text":"A030528 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A030528 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}