{"id":"n-012b9166110551b5","revision":0,"first_index":1,"fingerprint":"f6-fe1bba5e6d9fef5e","name":"mobius_x_over_x_plus_one of Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","9","9"],["1","24","66","44"],["1","55","320","530","265"],["1","118","1275","3940","4635","1854"],["1","245","4571","23485","48825","44499","14833"],["1","500","15400","123368","403270","628544","467236","133496"],["1","1011","49914","598416","2885694","6841674","8510376","5339844","1334961"],["1","2034","157785","2752320","18808230","63630588","117782490","121759560","66080565","14684570"],["1","4081","490589","12200595","114934380","532958118","1376892594","2085208950","1842913545","881074205","176214841"],["1","8176","1508430","52682564","670596135","4146081588","14358220800","29873769288","38227607055","29493959000","12599361126","2290792932"],["1","16367","4602884","223194686","3781378887","30552821013","137952719190","378683979960","657919707159","728281252985","498438362136","192426606294","32071101049"],["1","32750","13972231","932422764","20781541781","216131101186","1246972269543","4392153826632","9948681045303","14815428870242","14442017691101","8879686102924","3126932352271","481066515734"],["1","65517","42260463","3854868745","111986786275","1481544461397","10756736593603","47640591872445","136817064334665","263190154462235","342624926576933","298261239755115","166449014443985","53879449762215","7697064251745"],["1","131052","127501956","15812596528","594350595020","9908610918752","89458569672472","490588281660048","1750344923402790","4229525545919960","7060191650079568","8158753140966064","6414380565292700","3276704394169920","981375692097480","130850092279664"],["1","262123","384012830","64479935596","3117067154276","64987077217972","722660295348992","4848425588595160","21161153429488358","62934484848748370","130987011875542108","192926193870213440","200345803009008748","143575263753861380","67616785185516400","18842413288271624","2355301661033953"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["19","135","1171","11823","136459","1771815","25561891","405658143","7022891899","131714587095","2660335742611","57570797744463","1328913670495339","32592691757283975","846383665814211331"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","9","9"],["1","24","66","44"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","2","6","6","2"],["0","1","3","3","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","3","3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186881,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-954c16121d56b1d9"]},{"id":186882,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-954c16121d56b1d9"]},{"id":186883,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-954c16121d56b1d9"]}],"region_exact":{},"incoming_edges":[{"id":"e-954c16121d56b1d9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":2,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-012b9166110551b5","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":"2ab08acfc3aa02e8cc730f737254dec7cd30edb27a3f8a5bf247ee41ad71337f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-09153823987c7e65","role":"sequence","name":"Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-ed468c7343ff3515","kind":"fingerprint_related","node_a_id":"n-f85d6b4ccee7473c","node_b_id":"n-012b9166110551b5","fingerprint":"f6-fe1bba5e6d9fef5e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fe1bba5e6d9fef5e"},"other_id":"n-f85d6b4ccee7473c","other":{"id":"n-f85d6b4ccee7473c","name":"shift_x_plus_one of Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.","first_index":1,"fingerprint":"f6-fe1bba5e6d9fef5e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["9","9","1"],["44","66","24","1"],["265","530","320","55","1"]],"card_rows":[["9","9","1"],["44","66","24","1"],["265","530","320","55","1"],["1854","4635","3940","1275","118","1"]],"row_sums_preview":["19","135","1171","11823","136459","1771815","25561891","405658143"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-09153823987c7e65","steps":["x \u21a6 x+1"],"edges":["e-eb64db7fd8b24b13"],"text":"A046739 \u2192 x \u21a6 x+1"},"headline":"A046739 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1,    9,       9;\n   2: 1,   24,      66,       44;\n   3: 1,   55,     320,      530,       265;\n   4: 1,  118,    1275,     3940,      4635,       1854;\n   5: 1,  245,    4571,    23485,     48825,      44499,       14833;\n   6: 1,  500,   15400,   123368,    403270,     628544,      467236,      133496;\n   7: 1, 1011,   49914,   598416,   2885694,    6841674,     8510376,     5339844,     1334961;\n   8: 1, 2034,  157785,  2752320,  18808230,   63630588,   117782490,   121759560,    66080565,    14684570;\n   9: 1, 4081,  490589, 12200595, 114934380,  532958118,  1376892594,  2085208950,  1842913545,   881074205,   176214841;\n  10: 1, 8176, 1508430, 52682564, 670596135, 4146081588, 14358220800, 29873769288, 38227607055, 29493959000, 12599361126, 2290792932;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = ((n+2)*x+(3*n+6)*x^2+(3*n+6)*x^3+(n+2)*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 9*x + 9*x^2, P(2,x) = 1 + 24*x + 66*x^2 + 44*x^3.","mathematica":"P[1, x_] := 1 + 9*x + 9*x^2;\nP[2, x_] := 1 + 24*x + 66*x^2 + 44*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(((n + 2)*x + (3*n + 6)*x^2 + (3*n + 6)*x^3 + (n + 2)*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 9*x + 9*x^2; v[2] = 1 + 24*x + 66*x^2 + 44*x^3; for(n = 3, N, my(i = n); v[i] = (((n + 2)*x + (3*n + 6)*x^2 + (3*n + 6)*x^3 + (n + 2)*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-09153823987c7e65","steps":["x \u21a6 x/(x+1)"],"edges":["e-954c16121d56b1d9"],"text":"A046739 \u2192 x \u21a6 x/(x+1)"},"headline":"A046739 \u2192 x \u21a6 x/(x+1)","chain":null}
