{"id":"n-5638f00ca3f62529","revision":0,"first_index":0,"fingerprint":"f6-99a58a27f26df251","name":"mobius_x_over_x_plus_one of Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= 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":false,"rows":[["1"],["1","7"],["1","32","121"],["1","129","1515","3907"],["1","514","14766","103594","202741"],["1","2051","134914","1972486","9785429","15430207"],["1","8196","1212135","33595400","320622135","1226346036","1619195761"],["1","32773","10859997","547861985","9056469395","63493007871","197305208143","224061282907"],["1","131078","97379428","8784955586","239025234370","2752506013706","15176712147148","39681513963758","39531606447181"],["1","524295","874317564","140061381852","6105105409662","109191122431146","954855190814700","4327484068926684","9761297081388057","8661323866026007"],["1","2097160","7857847845","2230444161120","153497988983010","4129765159004352","54013027191077010","378693515297651520","1454755512281108445","2884453333240827160","2307185279184885001"],["1","8388617","70666104415","35537618724735","3831920621932650","152135600948101722","2874414182234504574","29352964381674735390","171223761049326335685","570271648278026118925","1008505284912214435571","734307168916191403507"],["1","33554442","635734892586","566763825905770","95396928088234995","5524004864871921492","147502206365287607484","2115493372145111426532","17624037421858706912295","87853509809430871983970","258101418121885003138266","411867922303844239590642","275199311597682485597221"],["1","134217739","5720406073350","9047239010246482","2373492576526837435","199038318330545534457","7404462620704407067908","145668227578115951552460","1669432611664439627575239","11706440133742133074335085","50883661291820480581022806","133669170036311392630504194","194309002966177267694253085","119956934012963778952439407"],["1","536870924","51478151732851","144526996646720104","59074063592488716641","7144493486829276985252","366800872361366497136523","9741434117988479998941552","149832414143295380233951083","1422570656409397347420883732","8594082738560572359583180001","33088344431942807012290157224","78581466720030847621171302451","104883094326613894061050492604","60172884133239122388455031841"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","154","5552","321616","27325088","3200979664","494474723072","97390246272256","23820397371219968","7083386168647642624","2516691244849530785792","1052914814802404260765696","512347915163742179541659648","286902390859642414913802102784"],"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","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["1","13","47","79","70","32","6"],["0","-3","-24","-63","-75","-42","-9"],["0","0","1","6","12","10","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","28","94","156","139","64","12"],["0","0","-2","-14","-36","-44","-26","-6"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","8","25","40","35","16","3"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193824,"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-19b1bfae8b1f232e"]},{"id":193825,"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-19b1bfae8b1f232e"]},{"id":193826,"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-19b1bfae8b1f232e"]}],"region_exact":{},"incoming_edges":[{"id":"e-19b1bfae8b1f232e","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-5638f00ca3f62529","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":"b888907c94c876a58209deb1d7ef41f97e95a9c7ae043c0225e2817b3b2caf25","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-4ef34abfaa61f0d6","role":"sequence","name":"Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= 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":[],"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,      7;\n  2: 1,     32,       121;\n  3: 1,    129,      1515,         3907;\n  4: 1,    514,     14766,       103594,        202741;\n  5: 1,   2051,    134914,      1972486,       9785429,        15430207;\n  6: 1,   8196,   1212135,     33595400,     320622135,      1226346036,      1619195761;\n  7: 1,  32773,  10859997,    547861985,    9056469395,     63493007871,    197305208143,     224061282907;\n  8: 1, 131078,  97379428,   8784955586,  239025234370,   2752506013706,  15176712147148,   39681513963758,   39531606447181;\n  9: 1, 524295, 874317564, 140061381852, 6105105409662, 109191122431146, 954855190814700, 4327484068926684, 9761297081388057, 8661323866026007;\n","recurrence_sentence":"The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(3*n-13)*x+(n^2-24*n+47)*x^2+(6*n^2-63*n+79)*x^3+(12*n^2-75*n+70)*x^4+(10*n^2-42*n+32)*x^5+(3*n^2-9*n+6)*x^6)*P(n-1,x) + (3*x-(2*n-28)*x^2-(14*n-94)*x^3-(36*n-156)*x^4-(44*n-139)*x^5-(26*n-64)*x^6-(6*n-12)*x^7)*d/dx P(n-1,x) + (x^2+8*x^3+25*x^4+40*x^5+35*x^6+16*x^7+3*x^8)*d^2/dx^2 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[((1 - (3*n - 13)*x + (n^2 - 24*n + 47)*x^2 + (6*n^2 - 63*n + 79)*x^3 + (12*n^2 - 75*n + 70)*x^4 + (10*n^2 - 42*n + 32)*x^5 + (3*n^2 - 9*n + 6)*x^6)*P[n - 1, x] + (3*x - (2*n - 28)*x^2 - (14*n - 94)*x^3 - (36*n - 156)*x^4 - (44*n - 139)*x^5 - (26*n - 64)*x^6 - (6*n - 12)*x^7)*D[P[n - 1, x], x] + (x^2 + 8*x^3 + 25*x^4 + 40*x^5 + 35*x^6 + 16*x^7 + 3*x^8)*D[P[n - 1, x], {x, 2}])/(1 + 3*x + 3*x^2 + 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] = ((1 - (3*n - 13)*x + (n^2 - 24*n + 47)*x^2 + (6*n^2 - 63*n + 79)*x^3 + (12*n^2 - 75*n + 70)*x^4 + (10*n^2 - 42*n + 32)*x^5 + (3*n^2 - 9*n + 6)*x^6)*v[i - 1] + (3*x - (2*n - 28)*x^2 - (14*n - 94)*x^3 - (36*n - 156)*x^4 - (44*n - 139)*x^5 - (26*n - 64)*x^6 - (6*n - 12)*x^7)*deriv(v[i - 1], x) + (x^2 + 8*x^3 + 25*x^4 + 40*x^5 + 35*x^6 + 16*x^7 + 3*x^8)*deriv(deriv(v[i - 1], x), x))/(1 + 3*x + 3*x^2 + 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-4ef34abfaa61f0d6","steps":["x \u21a6 x/(x+1)"],"edges":["e-19b1bfae8b1f232e"],"text":"A241171 \u2192 x \u21a6 x/(x+1)"},"headline":"A241171 \u2192 x \u21a6 x/(x+1)","chain":null}