{"id":"n-8617f30fe3d0f883","revision":1,"first_index":0,"fingerprint":"f6-511a49fff28ed5ac","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.","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":true,"rows":[["1"],["3","7"],["6","28","32"],["10","70","160","120"],["15","140","480","720","400"],["21","245","1120","2520","2800","1232"],["28","392","2240","6720","11200","9856","3584"],["36","588","4032","15120","33600","44352","32256","9984"],["45","840","6720","30240","84000","147840","161280","99840","26880"],["55","1155","10560","55440","184800","406560","591360","549120","295680","70400"],["66","1540","15840","95040","369600","975744","1774080","2196480","1774080","844800","180224"],["78","2002","22880","154440","686400","2114112","4612608","7138560","7687680","5491200","2342912","452608"],["91","2548","32032","240240","1201200","4228224","10762752","19987968","26906880","25625600","16400384","6336512","1118208"],["105","3185","43680","360360","2002000","7927920","23063040","49969920","80720640","96096000","82001920","47523840","16773120","2723840"],["120","3920","58240","524160","3203200","14094080","46126080","114216960","215255040","307507200","328007680","253460480","134184960","43581440","6553600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","66","360","1755","7938","34020","139968","557685","2165130","8227494","30705480","112842639","409209570","1466777160"],"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":[["0","0","0"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["2","11","18","11","2"],["1","2","-1","-4","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","5","9","7","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164223,"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-2eba49fde3ac73f5"]},{"id":164224,"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-2eba49fde3ac73f5"]},{"id":164225,"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-2eba49fde3ac73f5"]}],"region_exact":{},"incoming_edges":[{"id":"e-2eba49fde3ac73f5","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-8617f30fe3d0f883","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":"8b4af125d3a1eec2b1ad46181f76ad901b42676c7e60489283b799480764160a","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-32779d2065847504","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.","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":[{"id":"l-5b340e8309cd6b54","kind":"oeis_match","node_a_id":"n-8617f30fe3d0f883","node_b_id":null,"fingerprint":null,"oeis_id":"A080421","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A080421","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A080421","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-8617f30fe3d0f883","oeis_id":"A080421","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":11376}],"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,    7;\n  2:  6,   28,    32;\n  3: 10,   70,   160,   120;\n  4: 15,  140,   480,   720,    400;\n  5: 21,  245,  1120,  2520,   2800,   1232;\n  6: 28,  392,  2240,  6720,  11200,   9856,   3584;\n  7: 36,  588,  4032, 15120,  33600,  44352,  32256,   9984;\n  8: 45,  840,  6720, 30240,  84000, 147840, 161280,  99840,  26880;\n  9: 55, 1155, 10560, 55440, 184800, 406560, 591360, 549120, 295680, 70400;\n","recurrence_sentence":"The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (n+2+(2*n+11)*x-(n-18)*x^2-(4*n-11)*x^3-(2*n-2)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*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[((n + 2 + (2*n + 11)*x - (n - 18)*x^2 - (4*n - 11)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*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] = ((n + 2 + (2*n + 11)*x - (n - 18)*x^2 - (4*n - 11)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*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-c8c19eba9d68881f","steps":["(1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2","x \u21a6 x/(x+1)"],"edges":["e-6ba022abf95075ed","e-2eba49fde3ac73f5"],"text":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 x \u21a6 x/(x+1)","chain":null}