{"id":"n-45aa4f29cc8bfb73","revision":0,"first_index":0,"fingerprint":"f6-09bb967aa8554943","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["400","1680","1764"],["8000","50400","105840","74088"],["160000","1344000","4233600","5927040","3111696"],["3200000","33600000","141120000","296352000","311169600","130691232"],["64000000","806400000","4233600000","11854080000","18670176000","15682947840","5489031744"],["1280000000","18816000000","118540800000","414892800000","871274880000","1097806348800","768464444160","230539333248"],["25600000000","430080000000","3161088000000","13276569600000","34850995200000","58549671936000","61477155532800","36886293319680","9682651996416"],["512000000000","9676800000000","81285120000000","398297088000000","1254635827200000","2634735237120000","3688629331968000","3319766398771200","1742877359354880","406671383849472"],["10240000000000","215040000000000","2032128000000000","11379916800000000","41821194240000000","105389409484800000","184431466598400000","221317759918080000","174287735935488000","81334276769894400","17080198121677824"],["204800000000000","4730880000000000","49674240000000000","312947712000000000","1314380390400000000","3864278347776000000","8114984530329600000","12172476795494400000","12781100635269120000","8946770444688384000","3757643586769121280","717368321110468608"],["4096000000000000","103219200000000000","1192181760000000000","8345272320000000000","39431411712000000000","132489543352320000000","324599381213184000000","584278886183731200000","766866038116147200000","715741635575070720000","450917230412294553600","172168397066512465920","30129469486639681536"],["81920000000000000","2236416000000000000","28178841600000000000","216977080320000000000","1139129671680000000000","4305910158950400000000","12056548445061120000000","25318751734628352000000","39877033982039654400000","46523206312379596800000","39079493302398861312000","22381891618646620569600","7833662066526317199360","1265437718438866624512"],["1638400000000000000","48168960000000000000","657506304000000000000","5523052953600000000000","31895630807040000000000","133961649389568000000000","421979195577139200000000","1012750069385134080000000","1860928252495183872000000","2605299553493257420800000","2735564531167920291840000","2088976551073684586496000","1096712689313684407910400","354322561162882654863360","53148384174432398229504"],["32768000000000000000","1032192000000000000000","15173222400000000000000","138076323840000000000000","869880840192000000000000","4018849481687040000000000","14065973185904640000000000","37978127601942528000000000","79754067964079308800000000","130264977674662871040000000","164133871870075217510400000","156673241330526343987200000","109671268931368440791040000","53148384174432398229504000","15944515252329719468851200","2232232135326160725639168"],["655360000000000000000","22020096000000000000000","346816512000000000000000","3398801817600000000000000","23196822405120000000000000","116911984921804800000000000","450111141948948480000000000","1350333425846845440000000000","3190162718563172352000000000","5954970407984588390400000000","8753806499737344933888000000","10027087445153686015180800000","8773701514509475263283200000","5669160978606122477813760000","2551122440372755115016192000","714314283304371432204533760","93753749683698750476845056"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["3844","238328","14776336","916132832","56800235584","3521614606208","218340105584896","13537086546263552","839299365868340224","52036560683837093888","3226266762397899821056","200028539268669788905472","12401769434657526912139264","768909704948766668552634368","47672401706823533450263330816"],"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":[["400","1680","1764"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["20","62","42"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194867,"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-e125e216a3e2e975"]},{"id":194868,"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-e125e216a3e2e975"]},{"id":194869,"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-e125e216a3e2e975"]}],"region_exact":{},"incoming_edges":[{"id":"e-e125e216a3e2e975","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-45aa4f29cc8bfb73","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":"85d85928beded2ae8bcf95818f391dbfba68f3db2f60df0904133248cab966f3","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-7d2bbb701a5b14c4","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","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-47995ae1a4460723","kind":"fingerprint_related","node_a_id":"n-66d8297eef1ced69","node_b_id":"n-45aa4f29cc8bfb73","fingerprint":"f6-09bb967aa8554943","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-09bb967aa8554943"},"other_id":"n-66d8297eef1ced69","other":{"id":"n-66d8297eef1ced69","name":"basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","first_index":0,"fingerprint":"f6-09bb967aa8554943","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["21","10"],["441","420","100"]],"card_rows":[["1"],["21","10"],["441","420","100"],["9261","13230","6300","1000"]],"row_sums_preview":["1","31","961","29791","923521","28629151","887503681","27512614111"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e1d1088f7d1ff894","steps":["basis A007318"],"edges":["e-3706f9e3ae504ed3"],"text":"A038324 \u2192 basis A007318"},"headline":"A038324 \u2192 basis A007318","chain":null}},{"id":"l-7a899c60337081e2","kind":"fingerprint_related","node_a_id":"n-60cc7844ab862bfe","node_b_id":"n-45aa4f29cc8bfb73","fingerprint":"f6-09bb967aa8554943","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-09bb967aa8554943"},"other_id":"n-60cc7844ab862bfe","other":{"id":"n-60cc7844ab862bfe","name":"derivative of basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","first_index":0,"fingerprint":"f6-09bb967aa8554943","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["10"],["420","200"]],"card_rows":[["10"],["420","200"],["13230","12600","3000"],["370440","529200","252000","40000"]],"row_sums_preview":["0","10","620","28830","1191640","46176050","1717749060","62125257670"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e1d1088f7d1ff894","steps":["basis A007318","D"],"edges":["e-3706f9e3ae504ed3","e-2f60d9301140a587"],"text":"A038324 \u2192 basis A007318 \u2192 D"},"headline":"A038324 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-96d25b7bdbc4ad04","kind":"fingerprint_related","node_a_id":"n-45aa4f29cc8bfb73","node_b_id":"n-582aac7b79b62cfa","fingerprint":"f6-09bb967aa8554943","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-09bb967aa8554943"},"other_id":"n-582aac7b79b62cfa","other":{"id":"n-582aac7b79b62cfa","name":"basis_map_A013609 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-09bb967aa8554943","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["441","420","100"],["9261","13230","6300","1000"],["194481","370440","264600","84000","10000"]],"card_rows":[["441","420","100"],["9261","13230","6300","1000"],["194481","370440","264600","84000","10000"],["4084101","9724050","9261000","4410000","1050000","100000"]],"row_sums_preview":["961","29791","923521","28629151","887503681","27512614111","852891037441","26439622160671"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-2406b2ec50933490","e-6778caf68afabd2c"],"text":"A038319 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A038319 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-977c01425a5b1321","kind":"fingerprint_related","node_a_id":"n-0af4742945663751","node_b_id":"n-45aa4f29cc8bfb73","fingerprint":"f6-09bb967aa8554943","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-09bb967aa8554943"},"other_id":"n-0af4742945663751","other":{"id":"n-0af4742945663751","name":"reversal of basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","first_index":0,"fingerprint":"f6-09bb967aa8554943","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","21"],["100","420","441"]],"card_rows":[["1"],["10","21"],["100","420","441"],["1000","6300","13230","9261"]],"row_sums_preview":["1","31","961","29791","923521","28629151","887503681","27512614111"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e1d1088f7d1ff894","steps":["basis A007318","reverse"],"edges":["e-3706f9e3ae504ed3","e-f05dca09b09db3b3"],"text":"A038324 \u2192 basis A007318 \u2192 reverse"},"headline":"A038324 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-ed586b7e2ccb5b62","kind":"fingerprint_related","node_a_id":"n-45aa4f29cc8bfb73","node_b_id":"n-3c5688d9dc1ec135","fingerprint":"f6-09bb967aa8554943","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-09bb967aa8554943"},"other_id":"n-3c5688d9dc1ec135","other":{"id":"n-3c5688d9dc1ec135","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-09bb967aa8554943","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1764","1680","400"],["74088","105840","50400","8000"],["3111696","5927040","4233600","1344000","160000"]],"card_rows":[["1764","1680","400"],["74088","105840","50400","8000"],["3111696","5927040","4233600","1344000","160000"],["130691232","311169600","296352000","141120000","33600000","3200000"]],"row_sums_preview":["3844","238328","14776336","916132832","56800235584","3521614606208","218340105584896","13537086546263552"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["basis A013609","basis A007318"],"edges":["e-0c8cc5263f914382","e-39bd2e1b84fd7272"],"text":"A038336 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A038336 \u2192 basis A013609 \u2192 basis A007318","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:             400,             1680,              1764;\n  1:            8000,            50400,            105840,              74088;\n  2:          160000,          1344000,           4233600,            5927040,             3111696;\n  3:         3200000,         33600000,         141120000,          296352000,           311169600,           130691232;\n  4:        64000000,        806400000,        4233600000,        11854080000,         18670176000,         15682947840,          5489031744;\n  5:      1280000000,      18816000000,      118540800000,       414892800000,        871274880000,       1097806348800,        768464444160,         230539333248;\n  6:     25600000000,     430080000000,     3161088000000,     13276569600000,      34850995200000,      58549671936000,      61477155532800,       36886293319680,        9682651996416;\n  7:    512000000000,    9676800000000,    81285120000000,    398297088000000,    1254635827200000,    2634735237120000,    3688629331968000,     3319766398771200,     1742877359354880,     406671383849472;\n  8:  10240000000000,  215040000000000,  2032128000000000,  11379916800000000,   41821194240000000,  105389409484800000,  184431466598400000,   221317759918080000,   174287735935488000,   81334276769894400,   17080198121677824;\n  9: 204800000000000, 4730880000000000, 49674240000000000, 312947712000000000, 1314380390400000000, 3864278347776000000, 8114984530329600000, 12172476795494400000, 12781100635269120000, 8946770444688384000, 3757643586769121280, 717368321110468608;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (20+62*x+42*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1680*x + 1764*x^2.","mathematica":"P[0, x_] := 400 + 1680*x + 1764*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((20 + 62*x + 42*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 400 + 1680*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((20 + 62*x + 42*x^2)*v[i - 1])/(1 + x)); 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-cfc5b48d32898a3e","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-21c2803b3d0372c9","e-e125e216a3e2e975"],"text":"A038301 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038301 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}