{"id":"n-346eb8fc7de6d5a4","revision":0,"first_index":0,"fingerprint":"f6-a09e6eb68394f014","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.","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"],["1","2"],["3","11","10"],["12","64","113","66"],["55","385","1005","1159","498"],["273","2366","8164","14016","11969","4066"],["1428","14756","63280","144130","183855","124515","34970"],["7752","93024","476748","1352520","2293640","2324736","1303777","312066"],["43263","591261","3524367","11966409","25310565","34146819","28692153","13726991","2862562"],["246675","3782350","25704800","101612896","257471522","433625500","485366840","348148480","145198913","26824386"],["1430715","24322155","185598270","837116280","2471288820","4989225780","6975586464","6668728836","4171737339","1541911931","255680170"],["8414640","157073280","1329688425","6737989050","22707991800","53439422400","89604578700","107043573000","89280208500","49510870080","16429137361","2471150402"],["50067108","1018031196","9467451708","53245738260","201690506160","542062904526","1059858021222","1518936445026","1583529182235","1171110225285","583177567973","175560601111","24161357010"],["300830572","6618272584","67069384368","414531690496","1743321723220","5267828924424","11766237888768","19668626710656","24605051541147","22749743463370","15110540139772","6827373599424","1880753390593","238552980386"],["1822766520","43138807640","473148694720","3187627415544","14736366364148","49451232559300","124216068267720","237256311246240","346256816364888","384228602482296","319108186448480","192337746241030","79529837489895","20192670715859","2375085745978"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","24","255","3102","40854","566934","8164263","120864390","1827982362","28122626760","438720097638","6923868098820","110346550539780","1773394661610258"],"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","0"],["2","6","6","2"],["4","12","12","4"]]},"terms":[{"coeff":{"coeffs":[["6","32","68","72","38","8"],["-27","-143","-295","-297","-146","-28"],["27","141","281","267","120","20"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","19","47","57","34","8"],["0","-9","-55","-131","-153","-88","-20"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180870,"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-8e81a7ffdb9766a5"]},{"id":180871,"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-8e81a7ffdb9766a5"]},{"id":180872,"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-8e81a7ffdb9766a5"]}],"region_exact":{},"incoming_edges":[{"id":"e-8e81a7ffdb9766a5","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-346eb8fc7de6d5a4","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":"93cfa604450e9df3c47ee50218c33c6885c0e2af8066f5c6060d646dfcfd38a7","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-8e6b51ac707373f6","role":"sequence","name":"Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.","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-64ff0879d8922564","kind":"fingerprint_related","node_a_id":"n-346eb8fc7de6d5a4","node_b_id":"n-b662198f30afc0da","fingerprint":"f6-a09e6eb68394f014","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a09e6eb68394f014"},"other_id":"n-b662198f30afc0da","other":{"id":"n-b662198f30afc0da","name":"shift_x_plus_one of Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.","first_index":0,"fingerprint":"f6-a09e6eb68394f014","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["10","11","3"]],"card_rows":[["1"],["2","1"],["10","11","3"],["66","113","64","12"]],"row_sums_preview":["1","3","24","255","3102","40854","566934","8164263"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9ed87ca196739b1c","steps":["x \u21a6 x+1"],"edges":["e-c2f3c7ea34370dd0"],"text":"A104978 \u2192 x \u21a6 x+1"},"headline":"A104978 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-8cef7f557d10a9b7","kind":"fingerprint_related","node_a_id":"n-346eb8fc7de6d5a4","node_b_id":"n-6fe98fbfbcdc3b9a","fingerprint":"f6-a09e6eb68394f014","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a09e6eb68394f014"},"other_id":"n-6fe98fbfbcdc3b9a","other":{"id":"n-6fe98fbfbcdc3b9a","name":"shift_x_plus_one of Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Catalan triangle. T(n, k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-a09e6eb68394f014","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["30","33","9"]],"card_rows":[["1"],["4","2"],["30","33","9"],["264","452","256","48"]],"row_sums_preview":["1","6","72","1020","15510","245124","3968538","65314104"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-eb78d61c7c9e8c34","steps":["x \u21a6 x+1"],"edges":["e-b9e7ccb8b48c0ee8"],"text":"A330798 \u2192 x \u21a6 x+1"},"headline":"A330798 \u2192 x \u21a6 x+1","chain":null}}],"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:      1,       2;\n  2:      3,      11,       10;\n  3:     12,      64,      113,        66;\n  4:     55,     385,     1005,      1159,       498;\n  5:    273,    2366,     8164,     14016,     11969,      4066;\n  6:   1428,   14756,    63280,    144130,    183855,    124515,     34970;\n  7:   7752,   93024,   476748,   1352520,   2293640,   2324736,   1303777,    312066;\n  8:  43263,  591261,  3524367,  11966409,  25310565,  34146819,  28692153,  13726991,   2862562;\n  9: 246675, 3782350, 25704800, 101612896, 257471522, 433625500, 485366840, 348148480, 145198913, 26824386;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n+(12*n^2+6*n)*x+(12*n^2+6*n)*x^2+(4*n^2+2*n)*x^3)*P(n,x) = (27*n^2-27*n+6+(141*n^2-143*n+32)*x+(281*n^2-295*n+68)*x^2+(267*n^2-297*n+72)*x^3+(120*n^2-146*n+38)*x^4+(20*n^2-28*n+8)*x^5)*P(n-1,x) - ((9*n-3)*x+(55*n-19)*x^2+(131*n-47)*x^3+(153*n-57)*x^4+(88*n-34)*x^5+(20*n-8)*x^6)*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[((27*n^2 - 27*n + 6 + (141*n^2 - 143*n + 32)*x + (281*n^2 - 295*n + 68)*x^2 + (267*n^2 - 297*n + 72)*x^3 + (120*n^2 - 146*n + 38)*x^4 + (20*n^2 - 28*n + 8)*x^5)*P[n - 1, x] - ((9*n - 3)*x + (55*n - 19)*x^2 + (131*n - 47)*x^3 + (153*n - 57)*x^4 + (88*n - 34)*x^5 + (20*n - 8)*x^6)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (12*n^2 + 6*n)*x + (12*n^2 + 6*n)*x^2 + (4*n^2 + 2*n)*x^3)]];\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] = ((27*n^2 - 27*n + 6 + (141*n^2 - 143*n + 32)*x + (281*n^2 - 295*n + 68)*x^2 + (267*n^2 - 297*n + 72)*x^3 + (120*n^2 - 146*n + 38)*x^4 + (20*n^2 - 28*n + 8)*x^5)*v[i - 1] - ((9*n - 3)*x + (55*n - 19)*x^2 + (131*n - 47)*x^3 + (153*n - 57)*x^4 + (88*n - 34)*x^5 + (20*n - 8)*x^6)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (12*n^2 + 6*n)*x + (12*n^2 + 6*n)*x^2 + (4*n^2 + 2*n)*x^3)); 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-8e6b51ac707373f6","steps":["x \u21a6 x/(x+1)"],"edges":["e-8e81a7ffdb9766a5"],"text":"A108426 \u2192 x \u21a6 x/(x+1)"},"headline":"A108426 \u2192 x \u21a6 x/(x+1)","chain":null}