{"id":"n-5859809dd2683e24","revision":0,"first_index":0,"fingerprint":"f6-d02898a2b40396ec","name":"basis_map_A013610 of basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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":true,"rows":[["1"],["6","12"],["60","192","144"],["840","3408","4320","1728"],["15120","71424","118656","82944","20736"],["332640","1752768","3467520","3248640","1451520","248832"],["8648640","49563648","111188736","126074880","76723200","23887872","2985984"],["259459200","1590693120","3930425856","5116511232","3814594560","1637314560","376233984","35831808"],["8821612800","57197076480","152722796544","221126492160","191094349824","101443829760","32439730176","5733089280","429981696"],["335221286400","2279348259840","6489831186432","10235480260608","9915103199232","6147997728768","2450035703808","607134154752","85136375808","5159780352"],["14079294028800","99755282350080","299925088948224","507768145182720","539260097495040","377197142999040","176677472305152","54900062945280","10861337640960","1238347284480","61917364224"],["647647525324800","4757694516449280","14993617479819264","26956435745783808","30899182226964480","23822189747896320","12653529442025472","4645532563144704","1158422286827520","187300026777600","17708366168064","743008370688"],["32382376266240000","245656496126361600","806773208188354560","1527745197047021568","1868436340297629696","1561899674118389760","918542749076029440","384118981461540864","113667505099112448","23266068780810240","3133018629734400","249650812551168","8916100448256"],["1748648318376960000","13654039306018406400","46513631195687485440","92179519138799419392","119228504740636262400","106763818485964603392","68344104539526266880","31764937987835559936","10747473052890562560","2620377775353102336","448375831375380480","51077367434575872","3477279174819840","106993205379072"],["101421602465863680000","812918059569591091200","2861639081022095032320","5904575684398616150016","8021407504622496251904","7623043529073582145536","5245123885124098719744","2662495657768777678848","1004532692921679347712","280951587605166686208","57450331524009295872","8342997287709966336","814610601354461184","47932956009824256","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","396","10296","308880","10501920","399072960","16761064320","771008958720","38550447936000","2081724188544000","120740002935552000","7485880182004224000","494068092012278784000","34584766440859514880000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","12"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203613,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-e926a3c1d52c8fe4"]},{"id":203614,"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-e926a3c1d52c8fe4"]},{"id":203615,"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-e926a3c1d52c8fe4"]},{"id":203616,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-e926a3c1d52c8fe4"]}],"region_exact":{},"incoming_edges":[{"id":"e-e926a3c1d52c8fe4","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-5859809dd2683e24","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"b63b89bbedac16d414d23a513a46ccdad43be385786779caf1b3ab07091bdd81","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fb74a9e290f1102e","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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-4105b7e8b05c35fb","kind":"fingerprint_related","node_a_id":"n-ced07069205a749a","node_b_id":"n-5859809dd2683e24","fingerprint":"f6-d02898a2b40396ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d02898a2b40396ec"},"other_id":"n-ced07069205a749a","other":{"id":"n-ced07069205a749a","name":"reversal of basis_map_A013610 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","first_index":1,"fingerprint":"f6-d02898a2b40396ec","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["36","48","15"]],"card_rows":[["1"],["6","3"],["36","48","15"],["216","540","426","105"]],"row_sums_preview":["1","9","99","1287","19305","328185","6235515","130945815"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A013610","reverse"],"edges":["e-1b98e7733a40a4e4","e-16ee3a62654ccf95"],"text":"A161198 \u2192 basis A013610 \u2192 reverse"},"headline":"A161198 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-477cdb8ba374c6de","kind":"fingerprint_related","node_a_id":"n-5859809dd2683e24","node_b_id":"n-52a4cf9f8ec76487","fingerprint":"f6-d02898a2b40396ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d02898a2b40396ec"},"other_id":"n-52a4cf9f8ec76487","other":{"id":"n-52a4cf9f8ec76487","name":"basis_map_A013610 of basis_map_A013609 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","first_index":0,"fingerprint":"f6-d02898a2b40396ec","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["45","144","108"]],"card_rows":[["1"],["3","6"],["45","144","108"],["1575","6390","8100","3240"]],"row_sums_preview":["1","9","297","19305","2027025","310134825","64818178425","17695362710025"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A013609","basis A013610"],"edges":["e-2613ba1fbc2604f1","e-ae99e56a0fabb34b"],"text":"A204420 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A204420 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-4f873232d1de0d18","kind":"fingerprint_related","node_a_id":"n-fdb186db0412847e","node_b_id":"n-5859809dd2683e24","fingerprint":"f6-d02898a2b40396ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d02898a2b40396ec"},"other_id":"n-fdb186db0412847e","other":{"id":"n-fdb186db0412847e","name":"basis_map_A013610 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","first_index":1,"fingerprint":"f6-d02898a2b40396ec","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["15","48","36"]],"card_rows":[["1"],["3","6"],["15","48","36"],["105","426","540","216"]],"row_sums_preview":["1","9","99","1287","19305","328185","6235515","130945815"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A013610"],"edges":["e-1b98e7733a40a4e4"],"text":"A161198 \u2192 basis A013610"},"headline":"A161198 \u2192 basis A013610","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:            6,            12;\n  2:           60,           192,           144;\n  3:          840,          3408,          4320,           1728;\n  4:        15120,         71424,        118656,          82944,         20736;\n  5:       332640,       1752768,       3467520,        3248640,       1451520,        248832;\n  6:      8648640,      49563648,     111188736,      126074880,      76723200,      23887872,       2985984;\n  7:    259459200,    1590693120,    3930425856,     5116511232,    3814594560,    1637314560,     376233984,     35831808;\n  8:   8821612800,   57197076480,  152722796544,   221126492160,  191094349824,  101443829760,   32439730176,   5733089280,   429981696;\n  9: 335221286400, 2279348259840, 6489831186432, 10235480260608, 9915103199232, 6147997728768, 2450035703808, 607134154752, 85136375808, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+2+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n + 2 + 12*x)*P[n - 1, x]];\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] = (4*n + 2 + 12*x)*v[i - 1]); 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-4172472abe953cea","steps":["basis A013611","basis A013610"],"edges":["e-403803ced55fc794","e-e926a3c1d52c8fe4"],"text":"A290319 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A290319 \u2192 basis A013611 \u2192 basis A013610","chain":null}
