{"id":"n-d4b69b787803fc82","revision":0,"first_index":1,"fingerprint":"f6-56b745d77a7c0967","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","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"],["1","2"],["1","5","6"],["1","9","25","22"],["1","14","66","129","90"],["1","20","140","450","681","394"],["1","27","260","1210","2955","3653","1806"],["1","35","441","2765","9625","18963","19825","8558"],["1","44","700","5642","26110","72548","119812","108545","41586"],["1","54","1056","10584","62244","227808","526512","748548","598417","206098"],["1","65","1530","18600","134652","621180","1878240","3715260","4637205","3317445","1037718"],["1","77","2145","31020","269940","1521828","5764836","14840430","25653705","28537245","18474633","5293446"],["1","90","2926","49555","508860","3425928","15772680","50628798","113428095","174107010","174683718","103274625","27297738"],["1","104","3900","76362","911625","7199478","39385632","152989980","425769201","844160460","1165134828","1064611782","579168825","142078746"],["1","119","5096","114114","1564563","14287273","91264602","419584308","1408163757","3457368915","6146648508","7706630022","6464582943","3256957317","745387038"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","57","300","1686","9912","60213","374988","2381322","15361896","100389306","663180024","4421490924","29712558576"],"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"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","2","12","18","8"],["1","3","-1","-7","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","10","18","14","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175658,"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-ebfe679420eecb0e"]},{"id":175659,"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-ebfe679420eecb0e"]},{"id":175660,"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-ebfe679420eecb0e"]}],"region_exact":{},"incoming_edges":[{"id":"e-ebfe679420eecb0e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-d4b69b787803fc82","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":"5860e5c7da4f66f24db19a14ca57aba4d1460c16789e07400e1356b633d640e6","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-be392fc9b98fb771","role":"sequence","name":"Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","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-0b573f6eecd4ce8a","kind":"fingerprint_related","node_a_id":"n-d4b69b787803fc82","node_b_id":"n-3f9dac393051c6af","fingerprint":"f6-56b745d77a7c0967","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-56b745d77a7c0967"},"other_id":"n-3f9dac393051c6af","other":{"id":"n-3f9dac393051c6af","name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-56b745d77a7c0967","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["6","30","36"]],"card_rows":[["1"],["2","4"],["6","30","36"],["24","216","600","528"]],"row_sums_preview":["1","6","72","1368","36000","1213920","49956480","2427788160"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-694017a14c5c1f2e","steps":["x \u21a6 x/(x+1)"],"edges":["e-ce4a58ada30fbec2"],"text":"A253284 \u2192 x \u21a6 x/(x+1)"},"headline":"A253284 \u2192 x \u21a6 x/(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;\n   2: 1,  2;\n   3: 1,  5,    6;\n   4: 1,  9,   25,    22;\n   5: 1, 14,   66,   129,    90;\n   6: 1, 20,  140,   450,   681,    394;\n   7: 1, 27,  260,  1210,  2955,   3653,   1806;\n   8: 1, 35,  441,  2765,  9625,  18963,  19825,   8558;\n   9: 1, 44,  700,  5642, 26110,  72548, 119812, 108545,  41586;\n  10: 1, 54, 1056, 10584, 62244, 227808, 526512, 748548, 598417, 206098;\n","recurrence_sentence":"The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (n+(3*n+2)*x-(n-12)*x^2-(7*n-18)*x^3-(4*n-8)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n + (3*n + 2)*x - (n - 12)*x^2 - (7*n - 18)*x^3 - (4*n - 8)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n + (3*n + 2)*x - (n - 12)*x^2 - (7*n - 18)*x^3 - (4*n - 8)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-be392fc9b98fb771","steps":["x \u21a6 x/(x+1)"],"edges":["e-ebfe679420eecb0e"],"text":"A088617 \u2192 x \u21a6 x/(x+1)"},"headline":"A088617 \u2192 x \u21a6 x/(x+1)","chain":null}