{"id":"n-bca80792b78240c8","revision":0,"first_index":0,"fingerprint":"f6-d2b16266d15b8c0d","name":"mobius_x_over_x_plus_one of Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.","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"],["2","7","6"],["6","30","49","26"],["24","156","374","391","150"],["120","960","3030","4710","3601","1082"],["720","6840","26760","55140","63062","37927","9366"],["5040","55440","258720","663600","1009806","911190","451249","94586"],["40320","504000","2731680","8381520","15916824","19150236","14250854","5995591","1091670"],["362880","5080320","31358880","111978720","254850120","383251680","380739750","240899430","88073041","14174522"],["3628800","56246400","389491200","1586390400","4207699440","7592455080","9436983720","7976848980","4387780022","1418137447","204495126"],["39916800","678585600","5209142400","23830329600","72172734480","151918184880","226749048240","239949718800","176404548606","85799608950","24846302449","3245265146"],["479001600","8861529600","74684332800","379109808000","1290735008640","3104685460800","5409276269760","6877500632640","6332404740024","4117446611916","1794516768134","470675213191","56183135190"],["6227020800","124540416000","1143177235200","6375950380800","24104486165760","65220570455040","129924459624960","192916125642240","213490849692120","173898459134400","101334700955670","40002164242950","9587626273681","1053716696762"],["87178291200","1874333260800","18612565171200","113135627404800","470225173017600","1413534240518400","3169011259814400","5382204578150400","6958210045746960","6813830248144920","4974825728625480","2625875876316420","947200772224982","209000505036967","21282685940886"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","111","1095","13503","199815","3449631","68062695","1510769343","37260156615","1010843385951","29916558512295","959183053936383","33118910817665415"],"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":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","-1","-4","-5","-2"],["1","5","9","7","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":179377,"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-a314253597f2ac99"]},{"id":179378,"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-a314253597f2ac99"]},{"id":179379,"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-a314253597f2ac99"]}],"region_exact":{},"incoming_edges":[{"id":"e-a314253597f2ac99","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-bca80792b78240c8","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":"8a3b25bd0db0106c0fb501df380a0c415060a4c2c625e7975fabb4df56bc6ed1","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-45ec3349543b0d37","role":"sequence","name":"Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.","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-181119cd5c9b1d97","kind":"fingerprint_related","node_a_id":"n-bca80792b78240c8","node_b_id":"n-5d7f56cbbf85a498","fingerprint":"f6-d2b16266d15b8c0d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d2b16266d15b8c0d"},"other_id":"n-5d7f56cbbf85a498","other":{"id":"n-5d7f56cbbf85a498","name":"shift_x_plus_one of Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-d2b16266d15b8c0d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["6","7","2"]],"card_rows":[["1"],["2","1"],["6","7","2"],["26","49","30","6"]],"row_sums_preview":["1","3","15","111","1095","13503","199815","3449631"],"regions":["[-2, -1)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-677d34c7af372519","steps":["x \u21a6 x+1"],"edges":["e-493edac308631581"],"text":"A028246 \u2192 x \u21a6 x+1"},"headline":"A028246 \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:      2,       7,        6;\n  3:      6,      30,       49,        26;\n  4:     24,     156,      374,       391,       150;\n  5:    120,     960,     3030,      4710,      3601,      1082;\n  6:    720,    6840,    26760,     55140,     63062,     37927,      9366;\n  7:   5040,   55440,   258720,    663600,   1009806,    911190,    451249,     94586;\n  8:  40320,  504000,  2731680,   8381520,  15916824,  19150236,  14250854,   5995591,  1091670;\n  9: 362880, 5080320, 31358880, 111978720, 254850120, 383251680, 380739750, 240899430, 88073041, 14174522;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n+(5*n-1)*x+(9*n-4)*x^2+(7*n-5)*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 + (5*n - 1)*x + (9*n - 4)*x^2 + (7*n - 5)*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])/(1 + 2*x + 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 + (5*n - 1)*x + (9*n - 4)*x^2 + (7*n - 5)*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))/(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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-45ec3349543b0d37","steps":["x \u21a6 x/(x+1)"],"edges":["e-a314253597f2ac99"],"text":"A130850 \u2192 x \u21a6 x/(x+1)"},"headline":"A130850 \u2192 x \u21a6 x/(x+1)","chain":null}
