{"id":"n-59c900e21016c37e","revision":0,"first_index":0,"fingerprint":"f6-90334921cba1a1bb","name":"basis_map_A013610 of basis_map_A013609 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["9","36","36"],["81","648","1944","2592","1296"],["729","8748","43740","116640","174960","139968","46656"],["6561","104976","734832","2939328","7348320","11757312","11757312","6718464","1679616"],["59049","1180980","10628820","56687040","198404640","476171136","793618560","906992640","680244480","302330880","60466176"],["531441","12754584","140300424","935336160","4209012720","13468840704","31427294976","53875362816","67344203520","59861514240","35916908544","13060694016","2176782336"],["4782969","133923132","1741000716","13928005728","76604031504","306416126016","919248378048","2101139149824","3676993512192","4902658016256","4902658016256","3565569466368","1782784733184","548549148672","78364164096"],["43046721","1377495072","20662426080","192849310080","1253520515520","6016898474496","22061961073152","63034174494720","141826892613120","252136697978880","352991377170432","385081502367744","320901251973120","197477693521920","84633297223680","22568879259648","2821109907456"],["387420489","13947137604","237101339268","2529080952192","18968107141440","106221399992064","460292733298944","1578146514167808","4339902913961472","9644228697692160","17359611655845888","25250344226684928","29458734931132416","27192678397968384","19423341712834560","10359115580178432","3884668342566912","914039610015744","101559956668416"],["3486784401","139471376040","2649956144760","31799473737120","270295526765520","1729891371299328","8649456856496640","34597827425986560","112442939134456320","299847837691883520","659665242922143744","1199391350767534080","1799087026151301120","2214260955263139840","2214260955263139840","1771408764210511872","1107130477631569920","521002577708974080","173667525902991360","36561584400629760","3656158440062976"],["31381059609","1380766622796","28996099078716","386614654382880","3672839216637360","26444442359788992","149851840038804288","685036983034533888","2568888686379502080","7992098135402895360","20779455152047527936","45336993059012788224","83117820608190111744","127873570166446325760","164408875928288133120","175369467656840675328","153448284199735590912","108316435905695711232","60175797725386506240","25337177989636423680","7601153396890927104","1447838742264938496","131621703842267136"],["282429536481","13556617751088","311802208275024","4573099054700352","48017540074353696","384140320594829568","2432888697100587264","12511999013660163072","53175995808055693056","189070207317531353088","567210621952594059264","1443808855879330332672","3128252521071882387456","5775235423517321330688","9075369951241504948224","12100493268322006597632","13613054926862257422336","12812286989988006985728","9965112103324005433344","6293755012625687642112","3146877506312843821056","1198810478595369074688","326948312344191565824","56860576059859402752","4738381338321616896"],["2541865828329","132177023073108","3304425576827700","52870809229243200","608014306136296800","5350525893999411840","37453681257995882880","214021035759976473600","1016599919859888249600","4066399679439552998400","13825758910094480194560","40220389556638487838720","100550973891596219596800","216571328381899549900800","402203895566384878387200","643526232906215805419520","884848570246046732451840","1040998317936525567590400","1040998317936525567590400","876630162472863635865600","613641113731004545105920","350652064989145454346240","159387302267793388339200","55439061658362917683200","13859765414590729420800","2217562466334516707328","170581728179578208256"],["22876792454961","1281100377477816","34589710191901032","599554976659617888","7494437208245223600","71946597199154146560","551590578526848456960","3467140779311618872320","18202489091385999079680","80899951517271107020800","307419815765630206679040","1006101215232971585495040","2850620109826752825569280","7016911039573545416785920","15036237941943311607398400","28067644158294181667143680","45609921757228045209108480","64390477774910181471682560","78699472836001332909834240","82841550353685613589299200","74557395318317052230369280","56805634528241563604090880","36149040154335540475330560","18860368776175064595824640","7858486990072943581593600","2514715836823341946109952","580319039266925064486912","85973191002507416961024","6140942214464815497216"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","81","6561","531441","43046721","3486784401","282429536481","22876792454961","1853020188851841","150094635296999121","12157665459056928801","984770902183611232881","79766443076872509863361","6461081889226673298932241","523347633027360537213511521"],"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":[["9","36","36"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203120,"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-b5278d665d0d9315"]},{"id":203121,"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-b5278d665d0d9315"]},{"id":203122,"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-b5278d665d0d9315"]}],"region_exact":{},"incoming_edges":[{"id":"e-b5278d665d0d9315","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-59c900e21016c37e","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":"b02bff1e6c8181a5d3add47b002518e2eaff26abe959872bcf75f2e010acd840","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5c5dbc66494212e8","role":"sequence","name":"basis_map_A013609 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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-081b6db538d0c3ac","kind":"fingerprint_related","node_a_id":"n-4d8246302789a4f5","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-4d8246302789a4f5","other":{"id":"n-4d8246302789a4f5","name":"shift_x_plus_one of basis_map_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["32","32","8"],["1024","2048","1536","512","64"]],"card_rows":[["1"],["32","32","8"],["1024","2048","1536","512","64"],["32768","98304","122880","81920","30720","6144","512"]],"row_sums_preview":["1","72","5184","373248","26873856","1934917632","139314069504","10030613004288"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013609","basis A007318"],"edges":["e-65b163cdbb61d2ab","e-03a9f6536b0a9987"],"text":"A139548 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A139548 \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-093d7d694f3e98d6","kind":"fingerprint_related","node_a_id":"n-59c900e21016c37e","node_b_id":"n-3dda967a9c099e56","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-3dda967a9c099e56","other":{"id":"n-3dda967a9c099e56","name":"basis_map_A013610 of basis_map_A013611 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["36","144","144"],["1296","10368","31104","41472","20736"]],"card_rows":[["1"],["36","144","144"],["1296","10368","31104","41472","20736"],["46656","559872","2799360","7464960","11197440","8957952","2985984"]],"row_sums_preview":["1","324","104976","34012224","11019960576","3570467226624","1156831381426176","374813367582081024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013611","basis A013610"],"edges":["e-a6b5412d61e1c4e3","e-f1a89eb15ae97fc0"],"text":"A034870 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A034870 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-1162847e4ac0e336","kind":"fingerprint_related","node_a_id":"n-5c5dbc66494212e8","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-5c5dbc66494212e8","other":{"id":"n-5c5dbc66494212e8","name":"basis_map_A013609 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4","4"],["1","8","24","32","16"]],"card_rows":[["1"],["1","4","4"],["1","8","24","32","16"],["1","12","60","160","240","192","64"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A013609"],"edges":["e-20c0b01c4fb345ed"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013609"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013609","chain":null}},{"id":"l-1f3adf3c6b721c35","kind":"fingerprint_related","node_a_id":"n-a5e03e628d6f39e0","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-a5e03e628d6f39e0","other":{"id":"n-a5e03e628d6f39e0","name":"mobius_x_over_x_plus_one of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8","8"],["4","32","96","128","64"]],"card_rows":[["1"],["2","8","8"],["4","32","96","128","64"],["8","96","480","1280","1920","1536","512"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["x \u21a6 x/(x+1)"],"edges":["e-c45e154831ec4905"],"text":"A139548 \u2192 x \u21a6 x/(x+1)"},"headline":"A139548 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-63e8239c4dafea03","kind":"fingerprint_related","node_a_id":"n-8c0973a753fcbe28","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-8c0973a753fcbe28","other":{"id":"n-8c0973a753fcbe28","name":"basis_map_A007318 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4","1"],["16","32","24","8","1"]],"card_rows":[["1"],["4","4","1"],["16","32","24","8","1"],["64","192","240","160","60","12","1"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["x \u21a6 x+1"],"edges":["e-cfe8f3b6680af0c3"],"text":"A034870 \u2192 x \u21a6 x+1"},"headline":"A034870 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-708293438aea5c8e","kind":"fingerprint_related","node_a_id":"n-f8ceb867f43c1485","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-f8ceb867f43c1485","other":{"id":"n-f8ceb867f43c1485","name":"basis_map_A013611 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","32","32"],["64","512","1536","2048","1024"]],"card_rows":[["1"],["8","32","32"],["64","512","1536","2048","1024"],["512","6144","30720","81920","122880","98304","32768"]],"row_sums_preview":["1","72","5184","373248","26873856","1934917632","139314069504","10030613004288"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013611"],"edges":["e-770b6439e58a100f"],"text":"A139548 \u2192 basis A013611"},"headline":"A139548 \u2192 basis A013611","chain":null}},{"id":"l-7082e2929818d066","kind":"fingerprint_related","node_a_id":"n-d4fa80d8a6076549","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-d4fa80d8a6076549","other":{"id":"n-d4fa80d8a6076549","name":"shift_x_plus_one of basis_map_A013609 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","16","4"],["256","512","384","128","16"]],"card_rows":[["1"],["16","16","4"],["256","512","384","128","16"],["4096","12288","15360","10240","3840","768","64"]],"row_sums_preview":["1","36","1296","46656","1679616","60466176","2176782336","78364164096"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-9638204c7017296b","e-00495ecb8a6ad2ba"],"text":"A034870 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A034870 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-7739f96e805b5cc0","kind":"fingerprint_related","node_a_id":"n-0ca8dc857c80d449","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-0ca8dc857c80d449","other":{"id":"n-0ca8dc857c80d449","name":"basis_map_A013611 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16","16"],["16","128","384","512","256"]],"card_rows":[["1"],["4","16","16"],["16","128","384","512","256"],["64","768","3840","10240","15360","12288","4096"]],"row_sums_preview":["1","36","1296","46656","1679616","60466176","2176782336","78364164096"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013611"],"edges":["e-a6b5412d61e1c4e3"],"text":"A034870 \u2192 basis A013611"},"headline":"A034870 \u2192 basis A013611","chain":null}},{"id":"l-9b64f6d8790bd49f","kind":"fingerprint_related","node_a_id":"n-e289aa535cbb9f6a","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-e289aa535cbb9f6a","other":{"id":"n-e289aa535cbb9f6a","name":"basis_map_A013609 of theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","12","12"],["5","40","120","160","80"]],"card_rows":[["1"],["3","12","12"],["5","40","120","160","80"],["7","84","420","1120","1680","1344","448"]],"row_sums_preview":["1","27","405","5103","59049","649539","6908733","71744535"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["\u03b8+1","basis A013609"],"edges":["e-5ff024f1a04a7dc8","e-ca4c6b550df452a1"],"text":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1 \u2192 basis A013609"},"headline":"basis-map coefficients of A297191, shifted \u2192 \u03b8+1 \u2192 basis A013609","chain":null}},{"id":"l-a7d222ab6bb1f99c","kind":"fingerprint_related","node_a_id":"n-f16d27a1269a1da5","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-f16d27a1269a1da5","other":{"id":"n-f16d27a1269a1da5","name":"reversal of basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8","8"],["4","32","96","128","64"]],"card_rows":[["1"],["2","8","8"],["4","32","96","128","64"],["8","96","480","1280","1920","1536","512"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["x \u21a6 x+1","reverse"],"edges":["e-9d7e9cd87d9b797c","e-2067c388adcd5a73"],"text":"A139548 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A139548 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-b25dfda3487c528b","kind":"fingerprint_related","node_a_id":"n-4d3a36b3f07ad6a5","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-4d3a36b3f07ad6a5","other":{"id":"n-4d3a36b3f07ad6a5","name":"basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","8","2"],["64","128","96","32","4"]],"card_rows":[["1"],["8","8","2"],["64","128","96","32","4"],["512","1536","1920","1280","480","96","8"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["x \u21a6 x+1"],"edges":["e-9d7e9cd87d9b797c"],"text":"A139548 \u2192 x \u21a6 x+1"},"headline":"A139548 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-bf8e7bc04cfc0a0f","kind":"fingerprint_related","node_a_id":"n-d66b602a2fc4bda7","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-d66b602a2fc4bda7","other":{"id":"n-d66b602a2fc4bda7","name":"mobius_x_over_x_plus_one of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4","4"],["1","8","24","32","16"]],"card_rows":[["1"],["1","4","4"],["1","8","24","32","16"],["1","12","60","160","240","192","64"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["x \u21a6 x/(x+1)"],"edges":["e-78e2985691766434"],"text":"A034870 \u2192 x \u21a6 x/(x+1)"},"headline":"A034870 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f0a4cafce02083f5","kind":"fingerprint_related","node_a_id":"n-dc9add5d99860f64","node_b_id":"n-59c900e21016c37e","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-dc9add5d99860f64","other":{"id":"n-dc9add5d99860f64","name":"basis_map_A013609 of theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","36","36"],["25","200","600","800","400"]],"card_rows":[["1"],["9","36","36"],["25","200","600","800","400"],["49","588","2940","7840","11760","9408","3136"]],"row_sums_preview":["1","81","2025","35721","531441","7144929","89813529","1076168025"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["(\u03b8+1)\u00b2","basis A013609"],"edges":["e-5ac7ff393747d6fc","e-618052224ea3c031"],"text":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609"},"headline":"basis-map coefficients of A297191, shifted \u2192 (\u03b8+1)\u00b2 \u2192 basis A013609","chain":null}},{"id":"l-fd602551f23d6b69","kind":"fingerprint_related","node_a_id":"n-59c900e21016c37e","node_b_id":"n-6bdf4eb144359de4","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-6bdf4eb144359de4","other":{"id":"n-6bdf4eb144359de4","name":"basis_map_A013611 of basis_map_A013609 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","64","64"],["256","2048","6144","8192","4096"]],"card_rows":[["1"],["16","64","64"],["256","2048","6144","8192","4096"],["4096","49152","245760","655360","983040","786432","262144"]],"row_sums_preview":["1","144","20736","2985984","429981696","61917364224","8916100448256","1283918464548864"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013609","basis A013611"],"edges":["e-9638204c7017296b","e-cc31d0127172384e"],"text":"A034870 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A034870 \u2192 basis A013609 \u2192 basis A013611","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:         9,          36,           36;\n  2:        81,         648,         1944,          2592,           1296;\n  3:       729,        8748,        43740,        116640,         174960,          139968,           46656;\n  4:      6561,      104976,       734832,       2939328,        7348320,        11757312,        11757312,          6718464,          1679616;\n  5:     59049,     1180980,     10628820,      56687040,      198404640,       476171136,       793618560,        906992640,        680244480,        302330880,          60466176;\n  6:    531441,    12754584,    140300424,     935336160,     4209012720,     13468840704,     31427294976,      53875362816,      67344203520,      59861514240,       35916908544,       13060694016,        2176782336;\n  7:   4782969,   133923132,   1741000716,   13928005728,    76604031504,    306416126016,    919248378048,    2101139149824,    3676993512192,    4902658016256,     4902658016256,     3565569466368,     1782784733184,      548549148672,       78364164096;\n  8:  43046721,  1377495072,  20662426080,  192849310080,  1253520515520,   6016898474496,  22061961073152,   63034174494720,  141826892613120,  252136697978880,   352991377170432,   385081502367744,   320901251973120,   197477693521920,    84633297223680,    22568879259648,    2821109907456;\n  9: 387420489, 13947137604, 237101339268, 2529080952192, 18968107141440, 106221399992064, 460292733298944, 1578146514167808, 4339902913961472, 9644228697692160, 17359611655845888, 25250344226684928, 29458734931132416, 27192678397968384, 19423341712834560, 10359115580178432, 3884668342566912, 914039610015744, 101559956668416;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9+36*x+36*x^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[(9 + 36*x + 36*x^2)*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] = (9 + 36*x + 36*x^2)*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-cdb0db4ad473ee28","steps":["basis A013609","basis A013610"],"edges":["e-20c0b01c4fb345ed","e-b5278d665d0d9315"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013609 \u2192 basis A013610"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013609 \u2192 basis A013610","chain":null}